Monops , monoids and operads : The combinatorics of Sheffer polynomials .

We introduce a new algebraic construction, monop, that combines monoids (with respect to the product of species), and operads (monoids with respect to the substitution of species) in the same algebraic structure. By the use of properties of cancellative set-monops we construct a family of partially ordered sets whose prototypical examples are the Dowling lattices. They generalize the enriched partition posets associated to a cancellative operad, and the subset posets associated to a cancellative monoid. Their Whitney numbers of the first and second kind are the connecting coefficients of two umbral inverse Sheffer sequences with the family of powers {x}n=0. Equivalently, the entries of a Riordan matrix and its inverse. This aticle is the first part of a program in progress to develop a theory of Koszul duality for monops. Mathematics Subject Classifications: 05E99; 05A40, 06A07,06A11 Dedicated to the memory of Gian-Carlo Rota, 1932-1999.


Introduction
The systematic study of the Sheffer families of polynomials and of its particular instances: the Appel familes and the families of binomial type, was carried out by G.-C. Rota and his collaborators in what is called the Umbral Calculus (see [MR70,RKO73,RR78,Rom84]).A Sheffer sequence is uniquely associated to a pair of exponential formal power series, (F (x), G(x)), F (x) invertible with respect to the product of series, and G(x) with respect the electronic journal of combinatorics 25(3) (2018), #P3.25 to the substitution.The Sheffer sequences come in pairs, one is called the umbral inverse of the other.If one Sheffer sequence is associated to the pair (F (x), G(x)), its umbral inverse is associated to the pair ( 1 F (H(x)) , H(x)), where H(x) = F −1 (x), the substitutional inverse of F (x). Shapiro et al. introduced in [SGWC81] the Riordan group of matrices, whose entries in the exponential case, connect two Sheffer families of polynomials.Since that a great number of enumerative applications have been found by these methods.See for example the list of Riordan arrays of OEIS.
The initial motivation of the present research was to find a combinatorial explanation of the inversion process in the group of Riordan matrices.Equivalently, to the Sheffer sequences of polynomials and their umbral inverses.The key tool for such explanation is in the first place the concept of Möbius function and Möbius inversion over partially ordered sets (posets) [Rot64].
For the particular case of families of binomial type, the combinatorics of the process of inversion is related to families of posets of enriched partitions (assemblies of structures).One of the families of binomial type obtained by summation over the poset, and its umbral inverse by Möbius inversion.Particular cases of them were studied in [Rei78], [JRS81] and [Sag83].The general explanation was found in [MY91], where the construction of those posets is based on some special kind of set operads, called c-operads.
A similar approach can be applied to the Appel families.The central combinatorial object in this case is that of a c-monoid.A c-monoid is a special kind of monoid in the monoidal category of species with respect to the product (See [Joy81], [Men15].See also [AM10] for an extensive treatment of monoids and Hopf monoids in species).Given a c-monoid, through its product we are able to build a family of partially ordered sets.For each of these monoids, one Appel family is obtained by summation over those posets, and its umbral inverse by Möbius inversion.
In this article we introduce a new algebraic structure, that we call monop, because it is an interesting mix between monoids on species and operads.Our first step was to construct a monoidal category, the semidirect product (in the sense of Fuller [Ful16]) of the monoidal categories of species with respect to the product and the positive species with respect to the substitution.Then, we define a monop to be a pair of species (M, O) which is a monoid in such category.From the commutative diagrams satisfied for this kind of monoids we deduce all the main properties of monops, in particular that M is a monoid, O is an operad and M a right module over O satisfying natural compatibility properties.A special and particularly interesting class of monops is that of pairs of the form (O , O), an operad and its derivative.Giving the operadic product η : O(O) → O, by applying the derivative functor we get, by the chain rule By defining the monop product as ρ := η , we deduce the associativity property for monops from that of operads.In a similar way we obtain the monop unity and its properties.
We also introduce the c-monops.From a c-monop we give a general construction of posets that provide combinatorial explanation of the inverses of Riordan matrices by means of Möbius inversion.Or, equivalently, to Sheffer families and their umbral inverses.
We present a number of examples of Appel, binomial and general Sheffer families together with the posets constructed using the present approach.Remarkably, we obtain a new kind of operads, each of them associated to an arbitrary finite group.We call them the Dowling operads.By complementing the Dowling operads to monops we recover the classical Dowling lattice [Dow73].We also introduce here r-generalizations of the Dowling lattices for r a positive integer.With similar techniques we can define monops on rigid species (species over totally ordered sets), with the operations of ordinal product and substitution.In this way giving combinatorial interpretations to the inversion in the Riordan group associated to pairs of ordinary series (f (x), g(x)).
Posets associated to Monops have an independent algebraic interest beyond the enumerative applications given here.B. Vallette [Val07] proved that, under reasonable conditions, posets associate to a c-operad are Cohen-Macaulay [BGS82,Wac07] if and only if the c-operad is Koszul [GK94].In the same vein of Vallet approach, one of us has proved [Men10] that a c-monoid is Koszul if and only if the family of associated posets is Cohen-Macaulay.Our next step in this program shall be the development of a Koszul duality theory for monops.Monoids are closely related to associative algebras.The corresponding analytic functor M [Joy86] evaluated in a vector space is an associative algebra.The second component of the pair becomes a monad O, and M is a right-module over it.Then, Koszul duality for monops would establish a deep link between Koszul duality for operads and Koszul duality for associative algebras.And also, interesting connections with the Cohen-Macaulay property for the associated posets and Koszulness of the corresponding monop, unifying in this way the criteria established in [Val07] and in [Men10].

Formal power series
The exponential generating series (or function) of a sequence of numbers f n , n = 0, 1, . . . is the formal power series The coefficient f n will be denoted as F [n], F [n] := f n .The series F (x) will be called a delta series if F [0] = 0 and F [1] = 0.For an exponential series F (x) with zero constant term, F [0] = 0, we denote by γ k (F )(x) its divided power The substitution of such a formal power series F (x) in another arbitrary formal power series G(x) is equal to Riordan product of admissible pairs is defined as follows Admissible pairs of series in C[[x]] form a monoid with respect to the product * , having (1, x) as identity.The Riordan pairs form a group, the inverse of (F (x), G(x)) given by Where F −1 (x) and G −1 (x) denote the multiplicative and substitutional inverses of F (x) and G(x) respectively.
Definition 2. To an admissible pair (F (x), G(x)) we associate the infinite lower triangular matrix having as entries H k (x) being the series F (x).γ k (G(x)).That matrix is denoted as F (x), G(x) .The Riordan product is transported to matrix product by the bracket operator, We have that (see [SGWC81]).
The matrix F (x), G(x) is called a Riordan array when (F (x), G(x)) is a Riordan pair.Riordan arrays with the operation of matrix product form a group that is isomorphic to the group of Riordan pairs.The inverse of the matrix F (x), G(x) is equal to The ordinary generating function of the sequence f n is equal to the formal power series We denote by f [n] the nth.coefficient of f (x).
Definition 3.For an admissible pair (g(x), f (x)) of ordinary generating functions we define the associated matrix having as entries the coefficients where 3 Sheffer sequences of polynomials Definition 4. Let G(x) be a delta series.Define the polynomial sequence the electronic journal of combinatorics 25(3) (2018), #P3.25 and let p 0 (x) ≡ 1.This polynomial sequence is known to be of binomial type, It is called the conjugate sequence to the delta series G(x).It is also called the associated sequence to the series P (x) = G −1 (x).We have that where P (D) is the operator defined by D being the derivative operator Dr(x) = r (x).
Definition 5. We say that a family of polynomials s n (x) is Sheffer if there exists Riordan pair of formal power series (F (x), G(x)) such that By the Cauchy product formula we get Hence We will say that {s n (x)} ∞ n=0 is the conjugate sequence of (F (x), G(x)).Observe that the coefficients c n,k = F.γ k (G)[n] connecting the family of powers x n with s n (x), n 0, are the entries of the Riordan matrix associated to the pair (F (x), G(x)), F (x), G(x) .Let us consider the Riordan inverse of (F (x), G(x)), Let {p n (x)} ∞ n=0 be the family of binomial type associated to the delta operator P (D).We have the identity ) In effect, by Equations ( 8) and (11) the electronic journal of combinatorics 25(3) (2018), #P3.25 As a consequence of Eq. ( 13), we get that the Sheffer sequence {s n (x)} ∞ n=0 satisfies the binomial identity We say that it is Sheffer relative to the binomial family p n (x).It is called the Sheffer sequence associated to the Riordan pair (S(x), P (x)).A Sheffer sequence associated to a Riordan pair of the form (S(x), x) is called an Appel sequence.An Appel sequence is Sheffer relative to the family of powers, {x n } ∞ n=0 .Observe that, by Eq.( 10), a such Appel sequence a n (x) conjugate to the pair (F (x), x), F (x) = S −1 (x) is of the form, Similarly, a family of binomial type is Sheffer associated to Riordan pairs of the form (1, P (x)) (resp.conjugate to pairs of the form (1, F (x)), F (x) = P −1 (x).

Umbral substitution
Let be another Sheffer sequence conjugated to a Riordan pair (H(x), K(x)).Consider the umbral substitution defined by Since the matrix of coefficients of the umbral substitution is the product of the corresponding matrices, by Eq. (5), we have that Proposition 1.The umbral substitution s n (r) of two Sheffer sequences as above is also Sheffer, conjugated to the Riordan product Corollary 1.Let a n (x) and p n (x) be the Appel and binomial sequences conjugate respectively to (F (x), x) and (1, G(x)).Then we have Proof.Immediate from Prop. 1 and the identity the electronic journal of combinatorics 25(3) (2018), #P3.25 The Sheffer sequence associated to (F (x), G(x)) is the umbral inverse of {s n (x)} ∞ n=0 , denoted { s n (x)} ∞ n=0 .For every n 0,

Species and rigid species
In a general way, a (symmetric) species is a covariant functor from the category of finite sets and bijections B to a suitable category.For example, if we set as codomain the category of finite sets and functions F, we get set species (see [BLL98,Joy81]).If we instead set as codomain the category of vector spaces and linear maps Vec K ; we get linear species (see for example [Joy86,AM10,Men15]).By changing the domain B by the category of totally ordered sets L and poset isomorphisms, we obtain rigid species (species of structures without the action of the symmetric groups, non-symmetric species).Rigid species are endowed with two kinds of operations; shuffle and ordinal.
4.1 Three monoidal categories with species.
The (symmetric) set species, together with the natural transformation between them form a category.A species P is said to be positive if it assigns no structures to the empty set, P [∅] = ∅.The category of species will be denoted by Sp and the category of positive species by Sp + .
Recall that the product of species is defined as follows And the substitution of a positive species P into an arbitrary species R by The symbol of sum in set theoretical context will always denote disjoint union.The elements of the product M.N [V ] are pairs (m, n), m an element of M [V 1 ] and n an element the electronic journal of combinatorics 25(3 The category Sp is monoidal with respect to the operation of product.It has as identity the species 1 of empty sets, we have canonical isomorphisms The category of positive species is monoidal with respect to the operation of substitution.Its identity being the species of singletons, The divided power of an exponential formal power series G(x) has a counterpart in species.Recall that for a positive species P , The elements of γ k (P ) are assemblies of P -structures having exactly k elements, The elements of the substitution R(P ) are pairs of the form: (a, r), a = {p B } B∈π an assembly of P -structures, and r an element of R[π].The divided power can be seen as the substitution of P into the species E k , of sets of cardinal k, Definition 6.Let us consider now the product category Sp × Sp + .A pair of species (M, O) in Sp × Sp + will be called admissible.Morphisms are pairs of natural transformations of the form (φ, ψ) It is a monoidal category with respect to the Riordan product, defined as follows: having as identity the pair (1, X), It will be called from now on the Riordan category.
the electronic journal of combinatorics 25(3) (2018), #P3.25 The monoidal categories Sp and Sp + are respectively imbedded into the Riordan category by mapping, Remark 1.The Riordan category is just the semidirect product Sp Sp + (in the sense of [Ful16]) associated to the action The exponential generating functions of (M, O) is defined to be The generating function of the Riordan product (M 1 , O 1 ) * (M 2 , O 2 ) is obviously the Riordan product of the respective generating functions The matrix associated to an admissible pair Example 1.Let us consider E, the species of sets, E[V ] = {V }.Let E + be its associated positive species.The pair (E, E + ) has as generating function the Riordan pair The matrix associated to the pair (e x , e x − 1), Similar monoidal categories are defined on rigid species.Let R, S : L → F two rigid species.For l, a linear order on a set V , recall that the shuffle product and substitution are defined respectively by the electronic journal of combinatorics 25(3) (2018), #P3.25 where for V 1 ⊆ V , l V 1 denotes the restriction of the total order l to V 1 .Note that l induces a total order on any partition of V .We say that B < B , for B, B ∈ π if the minimun element of l B is smaller in l than the corresponding minimun element of l B .Applications of monops in the context of rigid species with ordinal product and substitution will be consider in a separated paper.

Cancellative monoids, cancellative operads, and posets.
A monoid in the monoidal category Sp, the species with the operation of product, is called (by language abuse) a monoid.An operad is a monoid in the category Sp + of positive species with respect to the substitution.More specifically.A monoid is a triplet (M, ν, e) such that the product ν : M • M → M is associative, and e : 1 → M , choses the identity, an element of M [∅].We also denote it by e, by abuse of language.We have then the associativity and identity properties .
and the pairs (e, m) and (m, e) respectively in And operad, as a monoid in Sp consists of a triplet (O, η, ν), where the product η : O(O) → O is associative, and for each unitary set {v}, e : X → O chooses the identity in denoted by e v .The product η sends pairs of the form ({ω B } B∈π , ω π ) into a bigger structure, ω V = η({ω B } B∈π , ω π ).Intuitively this product can be thought of as if η would assemble the pieces in a = {ω B } B∈π according to the external structure ω π .Associativity reads as follows, where a 2 is isomorphic to a 2 .By simplicity we will usually identify a 2 with a 2 .The identity property reads as follows See [Men15] for details and pictures.All this properties can be expressed by the commutativity of the diagrams of monoids in a monoidal category, see Section 7.
The plus sing meaning the disjoint union of the two graphs.There is another monoidal structure over G , the product ν 2 sending a pair of graphs to the graph obtained by connecting with edges all the vertices in g 1 with those in g 2 .The two monoidal structures are isomorphic by the correspondence c : g → g c , sending a graph to its complement, obtained by taking the complementary set of edges (with respect to the complete graph).
The natural transformation c is a monoid involutive isomorphism, c 2 = I G .The following diagram commutes (see also Fig. 1) The corresponding positive species G + is a c-operad with η({g B } B∈π , g π ) = g V , with g V as the graph obtained by keeping all the edges of the internal graphs plus some more edges created using the information of the external graph g π .For each external edge For a c-monoid (M, ν, e) we define a family of partially ordered sets where µ is the Möbius function of P M [V ].In a similar way, for a c-operad (O, η, e) we define a family of posets The elements of E + (O) are assemblies of O-structures.The order relation ν defined by where a 2 is an assembly with labels over the partition π 1 associated to a 1 , and having π 2 as associated partition, a 2 = { w D } D∈ π 2 .The product η defined as follows The poset P O [V ] has a zero, the assembly of singletons {e v } v∈V , e v the unique element of O[{v}].For M a c-monoid and O a c-operad, we define the Möbius generating functions of the respective family of posets We have that See [MY91,Men15].Moreover, we have.
Proposition 2. If we define the Appel and binomial families conjugated respectively to M (x) and O(x) the electronic journal of combinatorics 25(3) (2018), #P3.25 then, we have that their corresponding umbral inverses are obtained by Möbius inversion over the respective posets Proof.A proof of a more general proposition is given in Section 6, Theorem 3.

Examples of c-Monoids and Appel polynomials
Example 3. Pascal matrix, shifted powers.For the monoid E, P E [n] is the Boolean algebra of subsets of [n].The conjugate Appel is the shifted power sequence The umbral inverse obtained by Möbius inversion over P E [n] gives us their Appel umbral inverse Consider the power E r , the ballot monoid.The elements of E r [V ] are weak r compositions of V , i.e., r-uples of pairwise disjoint sets (V 1 , V 2 , . . ., V r ) (some of them possibly empty) whose union is V .It is a c-monoid by adding r-uples component to component: The ballot poset P E r [n] gives us the combinatorial interpretation of the umbral inversion between the Appel families (x + r) n and (x − r) n .
Example 4. Euler numbers The species of sets of even cardinal, E ev , is a submonoid of E. Its generating function is equal to the hyperbolic cosine, 2n! , (E * n being Euler or secant numbers, that count the number of zig permutations, OEIS A000364).We have that the electronic journal of combinatorics 25(3) (2018), #P3.25 The corresponding conjugate Appel polynomials are (OEIS A119467) and its umbral inverses (OEIS A119879) We have the identity (in umbral notation), And, making The classical Euler polynomials E n (x) are connected with a n (x) by the formulas The first identity follows by manipulating their generating functions, the second by binomial inversion.
Example 5. Free commutative monoid generated by a positive species.Let M be a positive species, the free commutative monoid generated by M is E(M ), the species of assemblies of M -structures.It is a c-monoid with the operation (a 1 , a 2 ) ν → a 1 + a 2 , taking the union of pairs of assemblies.The order in P E(M ) [n] is given by the subset relation on partial assemblies: The corresponding Appel polynomials are x |V | (37) Subsequent Examples 6, 7, and 8 are particular cases of this general construction.
the electronic journal of combinatorics 25(3) (2018), #P3.25 Example 6. Hermite Polinomials.Consider the free commutative monoid generated by E 2 , the species of sets of cardinal 2. It is the species of pairings.Equivalently, the species of partitions whose blocks all have cardinal 2, E(E 2 ), The elements of the poset P E(E 2 ) [n] are partial partitions of [n] having blocks of length two (partial pairings), endowed with the relation π 1 π 2 if every block of π 1 is a block of π 2 .The signless Hermite polynomials H n (x), are obtained as a sum over the elements of . Their umbral inverses, the Hermite polynomials H n (x) are obtained by Möbius inversion, see Fig. 2. In the figure, partial pairing are identified with a total partition having blocks of either size one or two.For example, following this convention, the partial partition of pairings 25|57 in {1, 2, 3, 4, 5, 6, 7}) is represented as a total partition 25|57|1|3|6 (also represented as the pair (25|57, {1, 3, 6})).In the following equations, π will represent a partial partition consisting only of pairings.
This elementary Möbius inversion is closely related to Rota-Wallstrom combinatorial approach to stochastic integrals for the case of a totally random Gaussian measure.See [RTW97], and [PT11].
Example 7. Bell-Appel polynomials.The free commutative monoid generated by E + , is equal to the species of partitions Π = E(E + ).The Bell-Appel polynomials conjugate to Π(x) = e e x −1 are The Möbius function of P Π [n] is equal to µ( 0, π) = (−1) |π| .Then, the umbral inverse B n (x) is equal to the electronic journal of combinatorics 25(3) (2018), #P3.25  ) , where k(G) is the number of connected components of G (The empty graph is assumed to have zero connected components).Their umbral inverses are the polynomials ) being the species of graphs having exactly j connected components.
Example 9.The species of lists L (totally ordered sets) is a c-monoid with product ν : L.L → L, the concatenation of lists.The poset P L [n] has as maximal elements the lists on [n].We have that l 1 l 2 if l 1 is an initial segment of l 2 .The Möbius function is as follows, the electronic journal of combinatorics 25(3) (2018), #P3.25 Their umbral inverses are

Examples of c-operads and binomial families
Example 10.The operad of lists and binomial Laguerre polynomials.The species of nonempty lists L + is an operad (the associative operad) with η the concatenation of lists following the order given by an external list, The elements of P L + [n] are linear partitions (partition with a total order on each block).
The coefficients counting such linear partitions having k blocks are the Lah numbers Hence, the polynomials obtained by summation on P L + [n] are the unsigned Laguerre polynomials (of binomial type) Since µ( 0, π) = (−1) n−|π| , by Möbius inversion we get that Example 11.Touchard polynomials.The operad E + gives rise to the poset of non-empty partitions ordered by refinement.The Touchard polynomials T n (x) conjugate to E + (x) = e x − 1 are Where S(n, k) are the Stirling numbers of the second kind.By Möbius inversion we obtain their umbral inverses It may be easily checked that e D − e −D 2 st n (x) = stn(x+1)−stn(x−1) 2 = nst n−1 (x).They are related to the Steffensen polynomials, [RR78], Ex. 6.1., by Example 13.The operad of cycles.Consider the rigid species of cyclic permutations C, A cyclic permutation can be identified with a linear order l having v 1 as first element, It is a shuffle c-operad with product η({l B } B∈π , l π ) = l B 1 l B i 2 . . .l B i k , the concatenation of the internal linear orders following the external order, l π = B 1 B i 2 . . .B i k .Since B 1 is the first element of the totally ordered set π = B 1 < B 2 < • • • < B k , the minimun element of B 1 is v 1 , and the product gives again a cyclic permutation.
The elements of the poset P C [n] are permutations (assemblies of cyclic permutations), hence the conjugate sequence of C(x) = ln( 1 1−x ) is the increasing factorial, If σ is a permutation with k cycles, the Möbius function was proved to be (see [JRS81]) µ( 0, σ) = (−1) n−k if all the cycles of σ are monotone, 0 otherwise.
Hence, their umbral inverses are T n (x) being Touchard polynomials.Example 14.The Abel sequence A n (x; a) = x(x + a) n−1 associated a xe −ax .For a = 1, it is conjugate to the generating series A (x) of rooted trees.The species of rooted trees has a c-operad structure (see [MY91]).In [Rei78] the poset P A [n] was constructed, and its Möbius function was computed in [Sag83].
Example 15.The Bessel polynomials of Krall and Frink y n (x).If we make K n (x) = x n y n−1 ( 1 x ) it is the associate sequence of x − x 2 2 .Hence, the conjugate to B(x), B being the species of commutative parethesizations, or commutative binary trees, satisfying the implicit equation It is the free operad generated by E 2 , a c-operad with the substitution of commutative parethesizations (or the grafting of commutative binary trees).Computing the inverse of where B n,k is the number of forests having k commutative binary trees with n labeled leaves.The Möbius function of such forests is µ( 0, a) = 0 if a has a tree with more than two leaves (−1) k k=number of binary trees with two leaves.
Their umbral inverse is the family K n (x), ), of the c-operad G c of connected graphs (Ex.2) has as conjugate the binomial family They are the generating function of graphs according to the number of their connected components.An explicit expression for their umbral inverses the electronic journal of combinatorics 25(3) (2018), #P3.25

The Dowling operad
Let G be a finite group of order m.Denote by E uG + the rigid species of G-colored ordered sets with an extra condition.The minimun element of the set is colored with the identity 1 of G.More explicitly, E uG + [∅] = ∅, and for a nonempty totally ordered set This kind of coloration will be called unital.It has as generating function This species has a structure of c-operad, η :

given as follows. The structures of E uG
+ (E uG + ) are pairs of the form ({f B } B∈π , g π ) where each f B is a unital coloration on B, and g π is a unital coloration on π (recall that π is a totally ordered set, B 1 < B 2 < • • • < B k , ordered according with their minimun element).The product h V = η({f B } B∈π , h π ) is obtained by multiplying by the right the "internal" colors on each block B ∈ π given by f B , times the "external" one given by h(B).Let b ∈ V , and B the unique block of π where it belongs.Then define h V (b) by where "•" is the product of the group.A unital coloration can be represented as a monomial with exponents on G.The elements of E uG + (E uG + )[V ] are then identified with factored monomials.This notation provides a better insight on the structure of the operad.
η : For example, for the multiplicative group of non-zero integers module 5, G = Z * 5 , and V = {a, b, c, d, e, f, g, h} we have: In Dowling's original setting of lattices associated to a finite group, he made use of equivalence classes of colorations over partial partitions of a set.If we had followed his approach this would have led us to the definition of an equivalence relation between G-colorations, f, h Observe that in each equivalence class of colorations there is only one which is unital.This is the reason why we define the Dowling operad by means of unital colorations.It is the natural way of avoiding complications with equivalence classes, by choosing one simple representative.Since G satisfies the left cancellation law, and we can define a posets ).The elements of the poset are assemblies of unital colorations, i.e., unital factored monomials.We say that a 1 a 2 if there exists a factored monomial a 2 over the factoras of a 1 such that η(a 1 , a 2 ) = a 2 .For example for G = Z * 5 , and naming A = a 1 b 2 , B = c 1 d 3 , C = e 1 f 2 g, and D = hc 2 and consider the factored monoid We have the product The poset P E uG + [n] has a unique minimal element 0, the assembly of trivial colorations over singletons, and m n−1 maximal elements (the number of unital G ions).The exponential generating function of the Möbius evaluation of is the substitutional inverse of the generating function The binomial family conjugate to Their umbral inverses being the electronic journal of combinatorics 25(3) (2018), #P3.25

Monops
At this stage, having studied two particular cases, what is missing is a a general construction of families of posets in order to give a combinatorial interpretation to the umbral inversion for Sheffer families.Or equivalently, to the inverses of Riordan arrays.To this end we define monoids in the Riordan category Sp Sp + .They will be called monops, because they are an interesting mix between a monoidal structure in the first component of the pair, with an operad structure in the second one.
Definition 7. A monop is a monoid in the Riordan category Sp Sp + .More specifically, an admissible pair of species (M, O) is called a monop if it is accompanied with a product (ρ, η), and identity morphisms (e, e), satisfying the identity and associativity properties of a monoid in the context of the Riordan category Sp Sp + .
We then have four natural transformations That suggest, without looking at the commuting diagrams implicit in the definition of (M, O), an operad structure on O, a monoid structure on M , and some extra conditions.We begin with two definitions in order to formulate those extra conditions.
Definition 8. Right module over an operad.
Let O be an operad and M a species.We say the M is a right module over O if we have an action τ : M (O) → M of O over M that is pseudo associative and where the assembly of identities of O fixes every structure of For a detailed study of modules over operads and applications see [Fre09].
Definition 9. Compatibility condition.Let (M, ν, e) be a monoid which is simultaneously a right module over O.We say that ν and τ are compatible if for every pair we have We postpone the proof of the Fundamental Theorem to Section 7.

Examples
Example 17.The pair (E, E + ) is a monop.The Boolean monoid E is a right module over the operad E + .There is a unique homomorphism τ V : It is easy to check that the module and monoid structure are compatible.Example 18.The pair (L, L + ) is a monop.The module structure is defined as follows.If V = ∅, τ ∅ is trivially defined.Otherwise we define τ a concatenation of linear orders as for the operad L + .The concatenation product ν : L.L → L is clearly compatible with τ .
Example 19.Let G c be the species of connected graphs.It is a c-operad with respect to the restriction of the product η defined in Ex. 2. The species of graphs is a c-monoid with respect to the product ν 1 of Ex. 2. It is also a right G c -module by restricting appropriately the product η of Ex. 2, to obtain τ : G (G c ) → G .It is easy to check that both structures are compatible.Hence the pair (G , G c ) is a c-monop.As a motivating example of the general procedure we will develop in Section 6, we are going to define a partial order over of graphs (an arbitrary graph and an assembly of connected graphs).The first element of the pair is called the monoidal section, and the assembly a the operadic section.We represent the pair (g 1 , {g B } B∈π ) by placing a double bar between the monoidal zone g 1 and the operadic one, and simple bars between the elements of the operadic zone {g B } B∈π (se Fig. 3).We say that (g if the assembly {g B } B∈π can be split in two subassemblies the electronic journal of combinatorics 25(3) (2018), #P3.25

{g
is less than or equal to {g C } C∈σ , in the partial order defined by the operad (G c , η).
In other words, a part of the assembly in the operadic zone of the pair is 'abducted' to the monoidal zone, and then transformed, by means of τ , in an element of the monoid.Finally it is multiplied, by means of ν 1 , with the element that initially was in the monoidal zone.
The other part of the assembly in the operadic zone, remains in it and then substituted by a bigger assembly (in the partial order defined by the operad).
We will give a general construction of posets of this kind obtained from c-monops.Each of them gives us a Sheffer family and their umbral inverses via Möbius inversion.

A family of monops: an operad and its derivative
For an operad O, the Riordan pair where ρ is the derivative of the morphism η : O(O) → O, ρ := η .In effect, by the chain rule we have that The pair (O , O) with the morphisms defined above is a monop (see Theorem 4 in Section 7 ).
Example 20.The structure of monop (E, E + ) in Ex. 17 can be defined by the derivative procedure, since E = E + .
6 Posets associated to c-monops Definition 10.A monop (M, O) is said to be a c-monop if O is a c-operad and M is a c-monoid and left cancellative as a right O-module.
For c-monop we will define a partially ordered set P (M,O) [V ].Recall that the subjacent set of the partially ordered set . By analogy we take the Riordan product with the monop (E, We already saw that for some decomposition of V as a disjoint union Before defining it we require the following definition of product. Definition 11.Let (m 1 , a 1 ) be an element of (M.E(O))[V ].Let π be the partition subjacent to the assembly a.Let (m 2 , a 2 ) be an element of Observe that either π 1 or π 2 may be empty.We define the product ρ((m 1 , a 1 ), (m 2 , a 2 )) := (ν(m 1 , τ (a 1 is the subassembly of a 1 having π i as subjacent partition, i = 1, 2. Observe that from the identity axioms for operads, monoids, and right Omodules we have that ρ((m, a), (e, {e B } B∈π )) = (a, m) = ρ((e, {e v } v∈V ), (m, a)), π being the partition subjacent to a.
Theorem 2. The product ρ is associative, left cancellative, and the identity does not have proper divisors.Let (m 1 , a 1 ), (m 2 , a 2 ) and (m 3 , a 3 ) be a triplet of nested elements of M.E(O), splitting of π, the partition subjacent to the assembly a 1 .

If
3. For (m 1 , a 1 ) an element of P (M,O) [V ], the order coideal Proof.Property 1 follows directly from Eq. ( 59) and Property 2 from the equivariance of ρ.To prove Property 3, choose an arbitrary element (m 2 , a 2 ) in P (M,O) [π] and define φ((m 2 , a 2 )) := ρ((m 1 , a 1 ), (m 2 , a 2 )).By the definition of the partial order, associativity, and the left cancellation law φ is an isomorphism.Property 4 is obtained in the same way by restricting φ to the interval [ 0, (m 2 , a 2 )].To prove Property 5, first observe that the product B∈π [ 0, {ω B }] B is isomorphic to the interval [ 0, a], a = {ω B } B∈π .Hence, we have to prove that the interval [ 0, (m, a)] is isomorphic to the product [ 0, (m, For an arbitrary element, ((m 1 , a 1 ), The last equation follows from Proposition 4 (properties 3 and 4), and the fact that if ρ((m 1 , a 1 ), (m 2 , a 2 )) = (m 2 , a 2 ), then Remark 2. Left divisibility on connected monoids in species, without the assumtion of the left cancellation law, is enough to define a partially ordered set (see [AM10], Section 8.7.6).However this property is essential for three reasons 1.Without the left cancellation law the partial order definition based on left divisibility can not be translated to other monoidal categories devoid of a grading and of a natural notion of connectedness.For example, in the category of ordinary set monoids it is not possible to define a partially ordered set by left divisibility only.3.In all the known monoidal categories in Species, a monoid satisfying the left cancellation law and the non-divisibility of the identity is connected.Perhaps the converse is also true, but we have not found a proof, nor a counterexample.
The elements of the partially ordered set P (E r ,E + ) [V ] are pairs (W, π), where W = (W 1 , W 2 , . . ., W r ) is a r-composition of some subset V 1 of V , and π is a partition of its complement in V .The partial order P (E r ,E + ) [V ] is better described by the covering relation.We say that (W, π) ≺ (W , π ) if either, 1.There exist a block B of π and some 1 i r, such that W = (W 1 , W 2 , . . ., W i + B, . . ., W r ), and π = π − {B}.
2. The partition π covers π in the refinement order, and W = W.
See Fig. 4 Their umbral inverses are the falling factorials the electronic journal of combinatorics 25(3) (2018), #P3.25 Example 23.Laguerre polynomials L [α] n (x).The Laguerre polynomials are Sheffer associated to ( 1 (1−x) α+1 , x x−1 ), For r = α + 1, a nonnegative integer, Let us consider the pair (L r , L + ).L r is the r-power of the monoid of lists, Ex. 9, and L + the operad of non-empty lists (the associative operad Ex. 10).It is a c-monop, L r a monoid with product ν the concatenation of r-uples of linear orders.It is also a compatible right L + -module with the action where l i is given by η being the product of the operad L + , and e the empty order, e ∈ L[∅].The elements of the poset , where l = (l 1 , l 2 , . . ., l r ) is an r-uple of linear orders and π is a linear partition.The numbers of such pairs satisfying |π| = k is easily proved to be n Hence, the Sheffer polynomials obtained by summation over P (L r ,L + ) [n] are The Möbius function is equal to By Möbius inversion, their umbral inverse family is equal to the electronic journal of combinatorics 25(3) (2018), #P3.25 Example 24.Poisson-Charlier polynomials.Consider the species of partitions Π.It is simultaneously the free commutative monoid generated by E + , Ex. 7, and the free right E + -module generated by E; Π = E(E + ) (see Ex. 7).As a free right E + module, the product is equal to τ = E(η), The monoid structure of Π is easily seen to be compatible with this module structure.Hence (Π, E + ) is a monop, more specifically, a c-monop.Its generating function and that of its inverse are the Riordan pairs Let (π 1 , π 2 ) and (π 3 , π 4 ) be two pairs of partitions in Π.Π[n].We will say that (π 1 , π 2 ) (π 3 , π 4 ) if we can split π 2 in two partitions, 2 , such that 1. π 3 = π 1 π 1 , π 1 being some partition on V 1 greater than or equal to π (1) 2 in the refinement order.
2. The partition π 4 covers π 2 in the refinement order of partitions.That is, π 4 is obtained by joining exactly two blocks of π 2 .
This family of posets gives us the combinatorics of the Poisson-Charlier polynomials and their umbral inverses.By summation over P (Π,E + ) [n] we get the Shifted Touchard polynomials T n (x + 1) By Möbius inversion we get the Poisson-Charlier polynomials corresponding to the parameter a = 1, c n (x; 1) = The general Poisson-Charlier polynomials c n (x; a) are the umbral inverses of the Sheffer family T n (ax + a), the electronic journal of combinatorics 25(3) (2018), #P3.25  τ : E r (E uG + ) → E r τ ({f B } B∈π , (π 1 , π 2 , . . ., π r )) = (∪ B∈π 1 B, ∪ B∈π 2 B, . . ., ∪ B∈πr B) where (π 1 , π 2 , . . ., π r ) is an r-composition of π, (π 1 , π 2 , . . ., π r ) ∈ E r [π].The reader may check that ν and τ are compatible.For r = 1 the pair (E, E uG + ) will be called the Dowling monop.In the next subsection we will give details of the construction of the Dowling and the r-Dowling posets.Observe that this example corresponds to the Riordan category in the context of L -species with shuffle product and substitution, their underlying sets are totally ordered.

The Dowling monop, Dowling lattices and the r-Dowling posets
The Dowling lattice Q n (G) is constructed using a monop (E, E uG + ).It has as underlying set (E.E(E uG + ))[{v 1 , v 2 , . . ., v n }], its elements are pairs of the form (V 1 , a), where a = {f B } B∈π is an assembly of unital colorations on V 2 , V = V 1 + V 2 .The partial order is defined as follows.
Definition 13.We will say that (V 1 , a 1 ) Qn(G) (V 3 , a 2 ) if the assembly a 1 splits in two subassemblies a 1 = a (1) 1 + a (2) 2 with respective underlying partitions π (1) and π (2) , such that G a 2 , where G is the partial order defined by the Dowling operad E uG + .
The order so defined is isomorphic to the classical Dowling lattice [Dow73].We are going to generalize this construction to a poset Q n,r (G) depending on a second parameter r and whose Whitney numbers of the first and second kind coincide with those defined in [MR17].The component α 2 is the associativity morphism in the category of positive species with respect to the substitution.The component α 1 is obtained by associativity with respect to the product of species, and then apply right hand side distibutivity of the substitution with respect to the product:

Figure 3 :
Figure 3: Definition of the partial order on G .E(G c )[V ] by means of the monop structure on (G , G c ).

2.
The left cancellation law allows interesting properties on the resulting poset, for example Property 3 in Proposition 4 above.Property 3 is responsible for the nice Möbius generating function inverse in Theorem 3. Analogous poset coideal properties and Möbius inversion generating function formulas are consequences of the left cancellation law in other contexts.For example, in the context of ordinary monoids, the classical inverse relation between the Dirchlet series of the Möbius function and the Riemann zeta function, associated to the monoid of positive integers.