Set partitions with colored singleton blocks

In this paper, we enumerate classes of partitions of [ n ] = { 1 , . . . , n } in which the singleton blocks are colored using a variable or ﬁxed number of colors. We consider, more generally, the distribution of the statistic recording the number of colored singletons on r -partitions of [ r + n ] in which only singletons from [ r + 1 , r + n ] may be colored. Among our results, it is shown by algebraic and bijective arguments that the number of partitions of [ n ] in which a singleton block { x } can come in one of x colors for each x is given by the n -th row sum of Lah numbers, yielding a new combinatorial interpretation for this sequence. Also, we show that the partitions of [ n ] in which each singleton is assigned one of s + 1 colors where s is ﬁxed are equinumerous with the set of s -partitions of [ s + n ] . Generalizations in terms of r -partitions of both of these results and others are demonstrated.


Introduction
By a partition of the set [n] = {1, . . ., n}, we mean a collection of nonempty disjoint subsets, called blocks, whose union is [n].Let P n denote the set of all partitions of [n].In this paper, we will deal with the enumeration of certain classes of P n wherein the singleton blocks are colored using either a fixed or a variable number of colors.There has been recent interest in enumerative combinatorics concerning statistics related to singleton blocks and counting classes of partitions of [n] (and other analogous structures) satisfying certain restrictions with regard to their singletons (see, e.g., [3,4,13,14,20,21] and references contained therein).For example, in [8,10,12], various classes of partitions are studied which contain no singletons.Perhaps some of the interest in the singletons statistic stems from the fact that its distribution on P n is the same as that of the parameter tracking the number of circular successions (i.e., occurrences of i and i + 1 in the same block of a partition, where n and 1 are regarded as consecutive, see, e.g., [7]).A combinatorial proof of this fact which makes use of an algorithmic bijection that switches singletons for block adjacencies and vice versa is given in [3].See [6] for other related statistics on set partitions.
By an r-partition of the set [r + n], we mean one in which the elements of [r] belong to distinct blocks.Let P (r) n denote the set of all r-partitions of [r + n].If the order matters in which the elements within each block in a member of P n , we mean one which contains an element of [r], with all other blocks being non-special.The same terminology will be applied at times to distinguish the elements themselves of [r] from those in n whose members contain exactly k non-special blocks (and hence r + k blocks altogether).Then the cardinality of L (r) n,k is given by the r-Lah number (see, e.g., [15,17]), which we will denote here by L n,k (see [16]), which gives the cardinality of all members of L (r) n,k .Note that when r = 0, the L (r) n,k reduce to the classical Lah numbers L n,k (see, e.g., [5] and entry A008297 in the OEIS [19]).The L (r) n reduce when r = 0 to the n-th row sum of Lah numbers given by [19].
Recall that the partial r-Bell polynomials (see, e.g., [11,18]), which are denoted by B (r) n,k (x i ; y i ) where x i and y i for i ≥ 1 are series of variables, are given explicitly by where Λ(n, k, r) is the set of all non-negative integer sequences (k i ) i≥1 and (r i ) i≥0 such that i≥1 k i = k, i≥0 r i = r and see [11,Corollary 4], which reduces to the well-known egf formula for B n,k (x i ) when r = 0.In what follows, we will make use of several particular cases of (2) in establishing our results.For example, the sequence of r-Lah numbers L (r) n,k coincides with the special case of B (r) n,k (x i ; y i ) where x i = y i = i! for all i.Hence, one has the egf formulas where the latter follows from the former by summing over all k ≥ 0.
In the next section, we introduce colored r-partitions of [r + n] where non-special singletons {x} come in one of x colors.We show that the number of such partitions is given by L (r) n , supplying both algebraic and combinatorial proofs.Thus, one gets a new combinatorial interpretation of this sequence and connection between r-partitions and r-Lah distributions.In the case when r = 0, one has a new interpretation of the sequence A000262 for the row sum of classical Lah numbers.We compute, more generally, the distribution of the statistic tracking the number of non-special singletons (marked by y) and show that it corresponds to a natural statistic on L (r) n .This equivalence is instrumental in explaining bijectively the y = −1 case of the distribution polynomial representing the sign balance of the singletons statistic.
A similar treatment is provided in the third section where non-special singletons in members of P (r) n are assigned one of a fixed number of colors.Among our results, we show that the r-partitions of [r + n] in which non-special singletons come in one of s + 1 colors are equinumerous with the set of (r + s)-partitions of [r + s + n].

Singletons with a variable number of colors
In this section, we consider colorings of set partitions in which a singleton {x} receives one of x possible colors.Before enumerating such partitions, recall that the (signless) r-Stirling number of the first kind (see, e.g., [2,15]) is defined by the recurrence c (r) Given n numbers i 1 , . . ., i n and 1 ≤ j ≤ n, define the j-th symmetric sum S j (i 1 , . . ., i n ) by To realize (4), we proceed inductively on n, noting that the equality holds if j = 0 or j which completes the induction and establishes (4).Let v n,k for n ≥ 0. Taking x i = 1 if i ≥ 2, with x 1 = 0, and y i = 1 for all i ≥ 1 in (2), and then summing over all k ≥ 0, implies that the egf for v (r) Let n denote the set of r-partitions of [r + n] in which a singleton {x} where x ∈ I comes in one of x colors.Given π ∈ K (r) n , let µ(π) denote the number of singletons in I. Define the distribution K Proof.Suppose that a member of K (r) n contains exactly j singleton blocks in I. Then there are S j {r + 1, . . ., r + n}y j possibilities for the (weighted) choice of these singletons, along with their colors.The remaining n − j members of I are then arranged together with the members of [r] according to an r-partition in v (r) n−j ways.Considering all possible 0 ≤ j ≤ n gives n,n−j y j , by (4), and formula (6) now follows by replacing j with n − j.
Define the egf for the sequence K (r) n (y) for n ≥ 0 and a fixed r by x n n! .
Theorem 2.1.We have Proof.By ( 6), ( 3) and ( 5), we have n≥0 K (r) n (y) Remark 2.1.The distribution of the colored singletons statistic on members of n having a fixed number k of non-special blocks is seen to be given by However, the egf over n ≥ k for a fixed k does not seem to have a simple formula since one cannot separate the v in the preceding proof).In the case of a fixed number of colors, it is possible to compute the analogous egf formula (see Lemma 3.1

below).
Taking y = 1 in the preceding result, and recalling n≥0 L n whose members contain an even or an odd number of singletons in I, respectively.Substituting y = −1 in Theorem 2.1 gives G (r) (x, −1) = e x (1+x) 2 , and extracting the coefficient of x n /n! yields the following sign-balance result.
i denote the n-th harmonic number for n ≥ 1.We have the following explicit formula for the total number of singletons in I among all members of K (r) n .
Theorem 2.2.If n ≥ 1 and r ≥ 0, then the total number of colored singletons in all r-partitions of [r +n] in which a singleton {x} where x ∈ [r + 1, r + n] is colored in one of x ways is given by Proof.Differentiating formula (7) with respect to y gives and hence Extracting the coefficient of x n n! for n ≥ 2, and recalling The two sums in the preceding expression may be combined to give which implies the desired formula.
From formula (6), it is seen that the total number of singletons in all the members of K (r) n is also given by the summation n,j .Equating this with the prior result gives the following apparently new identity relating r-Stirling, r-Lah and harmonic numbers.
Let t n denote the total number of non-special singletons in all the members of K (r) n for a fixed n ≥ 1 and r ≥ 0 variable.It is seen that t n is a polynomial in r of degree n.The first several t n are given in Table 1.Note that taking r = 0 in t n yields the total number of singletons in all the members of P n wherein each singleton {x} receives one of x colors.
We now provide combinatorial explanations of the formulas found above for the cardinality of K (r) n and for the sign balance of the colored singletons statistic on K (r) n .

Combinatorial proof of Corollary 2.1
We shall represent the element x in a colored singleton within a member of K n , we mean an element z in some block B of π in which all elements occurring to the right of z within B are greater than z.That is, if the (ordered) contents of B is given by the sequence i 1 • • • i p , then the letter i a corresponds to a block right-left minimum (rl min) if and only if i j > i a for all a < j ≤ p.
We first define a mapping from K (r) n , let P denote the set of singletons x y in π such that y ∈ [x − 1] and let P denote the partition of the members of [r + n] − P .If P is empty, then let f (π) = π, where we ignore the coloring of any singletons x x in π.So assume P is nonempty with the members of P given by i 1 < • • • < i for some ≥ 1.Let the singleton {i t } for t ∈ [ ] be assigned the color j t .We first insert i 1 into the partition P such that it directly precedes the element j 1 , but otherwise ignore the color assigned to i 1 .Next, we insert i 2 into the resulting (contents-ordered) partition such that it directly precedes j 2 , and proceed likewise with the elements i 3 , . . ., i , successively.Let f (π) denote the member of L (r) n that results once all the members of P have been inserted as described and the coloring of any singletons x x within π is disregarded.
Note that the inserted members of I from P correspond precisely to the set of elements that are not rl min within f (π).This follows from the fact that each element i t is inserted directly prior to a member of [r + n] that is less than i t and that such an insertion does not affect the status of any other rl min.Thus, the mapping f may be reversed as follows.If all the elements of σ ∈ L (r) n are rl min (i.e., if σ corresponds to a member of P (r) n ), then let g(σ) = σ, where any singletons {x} within σ are assigned the color x.So assume at least one element of [r + n] within σ does not correspond to an rl min and let a 1 < • • • < a denote the set of non rl min for some ≥ 1.First observe that {a 1 , . . ., a } ⊆ [r + 1, r + n] since members of [r] occur in different blocks of π.Further, we have that a must be followed directly by some member of [a − 1] within its block.For if not, then a not being the final element within any block implies it would be followed by some z > a .But then a being the largest non rl min implies z must be an rl min.Thus, all of the elements to the right of z in its block must be greater than z, and hence a .This would imply a would be an rl min, which it isn't.
We then remove a from its block within σ and form the singleton {a }, which we assign the color q, where q ∈ [a − 1] denotes the successor of a within its block.Note that moving a as described does not create or destroy any rl min (in blocks other than the new singleton {a }), as the successor of a in its block was smaller than a .In the resulting partition where a occurs as a colored singleton, we have that a −1 must be followed by a smaller element t in its block, upon reasoning as before.We then move a −1 from its block and create the singleton {a −1 }, which is assigned the color t.We proceed likewise, successively, with a −2 , . . ., a 1 .After a 1 has been moved and its singleton assigned some color in [a 1 − 1], we assign to any uncolored singletons {x} where x ∈ I occurring within the partition at this point the color x.Let g(σ) denote the resulting member of K (r) then we have f (π) = σ and g(σ) = π.One can then show g(f (π)) = π for all π ∈ K (r) n since g when applied to f (π) is seen to restore each of the singleton blocks of π in I along with their respective colors.Likewise, f (g(σ)) = σ for all σ ∈ L (r) n since f when applied to g(σ) sequentially recovers the non rl min of σ in the reverse order in which they were taken away by g.Thus, the mapping f provides a bijection between K (r)

Combinatorial proof of Corollary 2.2
We first show where d n denotes the number of derangements of [n] (i.e., permutations without fixed points), see A000166 in [19].We demonstrate first the r = 0 case of ( 9), where we may clearly assume n ≥ 2. From the combinatorial proof of Corollary 2.1 above, one has that the statistic on n recording the number of singleton blocks has that same distribution as the statistic on L n = L (0) n which records the number of elements p ∈ [n] within π ∈ L n such that either (i) p is not a block rl min of π or (ii) p is both the last and smallest element of some block of π.Define the sign of π by (−1) σ(π) , where σ(π) denotes the number of elements of [n] satisfying either condition (i) or (ii).It then suffices to define a sign-changing involution on L n off of a subset of L n whose members have sum of signs given by the right-hand side of (9).
To do so, we first order the blocks of π ∈ L n from left to right in increasing order of their smallest elements contained therein.Consider a block B of π, if it exists, such that |B| ≥ 2 whose last element (when considering the sequence of elements within B from left to right) is neither the smallest nor the second smallest element of B. Assume B is the leftmost block of π satisfying this requirement and let a, b with a < b denote the two smallest elements of B. We then switch the elements a and b within B, leaving all other members of B undisturbed.Let υ(π) denote the resulting member of L n .Note that b changes its status concerning being an rl min of π, with no other members of [n] changing in this regard.Further, since neither a nor b is the final element of B, we have that π and υ(π) are of opposite σ-parity.Thus, υ yields a sign-changing involution of L n in all cases when it is defined.Note that if a were to occur at the end of B, then switching a and b would fail to reverse the parity since a and b would both go from contributing to the σ value of π to neither doing so, with all other elements of [n] remaining of the same status concerning their contributing to σ(π).
We now determine the sum of the signs of the members of the set T of survivors of υ, which is seen to consist of those π in which each non-singleton block B ends either in its smallest or second smallest element.Note that all elements in singleton blocks contribute to the value of σ(π), with the same holding for non-singletons whose smallest element is last.If B is a non-singleton block whose second smallest letter is last, then the smallest two elements of that block do not contribute to σ(π), with all other elements in B doing so (being non rl min).Thus, it is seen that each member of T has sign (−1) n .To enumerate the members of T , let m denote the number of elements of [n] going in either singletons or non-singletons whose smallest element is last.Once those members of [n] have been selected, there are m! ways in which to arrange them (as blocks in this case may be viewed as cycles of an arbitrary permutation of size m), with d n−m ways in which to arrange the unselected members of [n] as the remaining blocks are required to be non-singletons in which the second smallest element is last.Considering all possible 0 ≤ m ≤ n implies that the sum of the signs of the members of T is given by the right side of (9), which completes the proof in the r = 0 case.We now show (9) in the case when r > 0. Recall that a special block within a member of L n is one that contains a special element, i.e., a member of [r].Note that from the proof of Corollary 2.1, we have that the colored singletons statistic on K (r) n where r > 0 corresponds to the statistic on L (r) n recording the number of x ∈ I such that one of the following holds: (a) x occurs as a non rl min within a non-special block, (b) x occurs as both the smallest and last element of some non-special block, (c) x occurs as a non rl min in the sequence of elements to the right of the special element within some special block or (d) x occurs anywhere to the left of the special element within a special block.Let δ(ρ) denote the number of elements of I satisfying one of the conditions (a)-(d) above within ρ ∈ L (r) n .If the sign of ρ is defined as (−1) δ(ρ) for each ρ, then by the preceding the sum of signs of all members of L (r) n is given by the left-hand side of (9).We now define an involution of L (r) n which reverses the δ-parity.If no elements of I occur in the special blocks of ρ ∈ L (r) n , then we may apply the mapping υ defined above since only conditions (a) and (b) would apply.Otherwise, let s denote the smallest member of [r] such that the s-th special block of ρ contains at least one member of I. Suppose first that at least two members of I occur to the right of s within its block S and let u < v denote the smallest two such members.Then switching the positions of u and v is seen to change the parity since only the element v changes in regard to its contributing to the value of δ (as its status concerning condition (c) changes).Note here it is possible for either u or v to occur at the very end of S since u would not be counted by δ, regardless of its position.Now suppose there is at most one element of I occurring to the right of s within S. Let w denote the rightmost element of I occurring in S in this case.We then switch the letters s and w and observe that this changes the status of w with regard to its satisfying (d) above.Note further that w would not satisfy (c) when it occurs last in S since it would trivially be an rl min in this case.Therefore, each member of L (r) n for which at least one element of I occurs within a special block is paired with another of opposite δ-parity, which implies formula ( 9) holds for all r > 0 as well.
the order of the blocks themselves being immaterial), one obtains what is known as an r-Lah distribution.Let L (r) n denote the set of all r-Lah distributions of [r + n].By a special block within a member of P (r) n,k and is given explicitly by n! k! n+2r−1 k+2r−1 for all non-negative n, k and r.Let L (r) n,k denote the number of r-partitions of [r + n] into r + k blocks in which no singleton of the form {z}, where z ∈ I occurs.Define v (r) terms when sums are interchanged (as happens with v

1 ( 1
−x) 2r exp x 1−x , yields the following connection between colored set partitions and Lah distributions.Corollary 2.1.If n, r ≥ 0, then |K (r) n | = L (r) n .In particular, the number of partitions of [n] in which a singleton {x} is colored in one of x ways for each x ∈ [n] is given by L n .Let E (r) n and O (r) n denote the subsets of K (r)