Eulerian Numbers Associated with Arithmetical Progressions

In this paper, we give a combinatorial interpretation of the r-Whitney-Eulerian numbers by means of coloured signed permutations. This sequence is a generalization of the well-known Eulerian numbers and it is connected to r-Whitney numbers of the second kind. Using generating functions, we provide some combinatorial identities and the log-concavity property. Finally, we show some basic congruences involving the r-Whitney-Eulerian numbers.


Introduction
The Eulerian numbers were introduced by Euler in a noncombinatorial way.Euler was trying to obtain a formula for the alternating sum m i=1 i n (−1) i (cf.[10]).Explicitly, Eulerian numbers A(n, k) can be defined by the recurrence relation [6] the electronic journal of combinatorics 25(1) (2018), #P1.48 with the initial values A(n, 1) = 1 for n 0 and A(0, k) = 0 if k 2. Eulerian numbers can also be computed by the following expression where S(n, m) are the Stirling numbers of the second kind.
Another interesting identity involving Eulerian numbers is called Worpitzky's identity It is well-known that Eulerian numbers have a combinatorial interpretation in term of permutations.In particular, the Eulerian number A(n, k) counts the number of permu- The Eulerian polynomials are defined by with A 0 (x) = 1.These polynomials satisfy the following relation for any non-negative integer n [6, p. 245].
The Eulerian numbers and their generalizations have been studied extensively (cf.[21]).In the present article, we are interested in a recent generalization called r-Whitney-Eulerian numbers and denoted by A m,r (n, k) in [19].This new sequence is defined by the expression where W m,r (n, k) are the r-Whitney numbers of the second kind.The r-Whitney numbers of the second kind W m,r (n, k) were defined by Mező [16] as the connecting coefficients between some special polynomials.Specifically, for non-negative integers n, k and r with n k 0 and for any integer m > 0 where The r-Whitney numbers of the second kind satisfy the recurrence [16] W Note that if (m, r) = (1, 0) we obtain the Stirling numbers of the second kind, if (m, r) = (1, r) we have the r-Stirling (or noncentral Stirling) numbers [4], and if (m, r) = (m, 1) we have the Whitney numbers [1].For more details on r-Whitney numbers see for example [5,7,14,15,18,20,23,24].
The r-Whitney-Eulerian polynomials are defined by For non-negative integers r, n and positive m, it is known [19] that they satisfy the following identity and their exponential generating function is For a similar class of Eulerian numbers connected to the Whitney numbers see the papers of Rahmani [22] and Mező [17].
In the present article, we give a combinatorial interpretation of the r-Whitney-Eulerian numbers by means of coloured signed permutations.Afterwards, we find several combinatorial identities in terms of this new sequence.Moreover, we prove that the r-Whitney-Eulerian numbers are log-concave and therefore unimodal.Finally, we establish some interesting congruences involving this sequence.
A signed permutation σ ∈ B n is (m, r)-coloured if it satisfies the following conditions: ∈ Inv B (σ) for all i, and σ(i) > 0 then σ(i) is coloured with one of r colors.But, if σ(i) < 0 then it is coloured with one of m − r colours.
• If the above inversion property does not hold, then we colour σ(i) with one of m − 1 colours providing that σ(i) < 0, but if σ(i) is positive we coloured it with one colour.
Let n, k, m, r 0 be integers with m r.Let B denote the set of (m, r)-coloured signed permutations of B n with k descents.
Theorem 1.For any integers n, k, m, r 0, with m r we have In the first case, we have to put the entry n at the end of π , or we have to put the entries n or −n between two entries that form one of the k descents of π .Then we have the following possibilities: In the second case, we have to put the entries n or −n at the beginning of π , or we have to put the entry −n at the end of π or we have to insert n or −n between one of the (n − 2) − (k − 1) = n − k − 1 ascents of π .Hence we have the following possibilities Example 2. Let n = 2, m = 3 and r = 2.The m − 1 = 2 different colours of the elements will be fixed as red and green; the r = 2 different colours of the elements will be fixed as cyan and blue; while the m − r = 1 colours of the elements will be fixed as magenta.Therefore, A 3,2 (2, 0) = 4, A 3,2 (2, 1) = 13 and A 3,2 (2, 2) = 1, where the coloured signed permutations are in Table 1.
Theorem 3. The following identity holds ∈ Inv(|σ|) for any j > i}.We suppose that = |P σ |, and suppose there are t negative positions of these (0 t ), then these negative positions can be coloured with one of m − r colours, while the − t positive positions can be coloured with one of r-colours.Therefore by the product rule we have ways to colour each fixed permutation.So, summing over all possible non-signed permutations we get the desired identity.

Some Combinatorial Identities
The goal of the current section is to extend some well-known identities for the classical Eulerian numbers to the r-Whitney-Eulerian numbers.
Theorem 4. For n, k 0, we have the following identity Proof.The proof follows by showing that the right side of the identity have the same recurrence relations as the r-Whitney numbers of the second kind.
the electronic journal of combinatorics 25(1) (2018), #P1.48 The r-Whitney-Eulerian numbers are not symmetric as the classical Eulerian numbers (A(n, k) = A(n, n − k + 1)).However, we note that A m,r (n, n − k − 1) = A m,r (n, k), where A m,r (n, k) are the generalized Eulerian numbers defined by Xiong et al. [25].From above relation and Lemmas 7 and 8 of [25], we obtain a generalization of the Worpitzky's identity.Theorem 5.For n 0, we have the identities Theorem 6 gives a generalization of the well-known identity for the Eulerian numbers (cf.[6, p. 243]) Theorem 6.For n, k 0, we have the identity Proof.By using the generating function (7) we have Comparing the coefficients on both sides, we get the desired result.
Above identity gives us special values when k is small: Finally, by using the generating function (7) we find a relation between the r-Whitney-Eulerian polynomials and the classical Eulerian polynomials.Theorem 7.For n 0, we have the following identity

Unimodality and Log-Concavity Properties
In this section we prove the log-concavity and therefore the unimodality of the r-Whitney-Eulerian numbers.Recall that a finite sequence of non negative real numbers {a k } 0 k n is said to be unimodal if there is an index i such that a 0 a It is well know that a sequence which is log-concave is also unimodal.We first prove the following equality.Theorem 8.For n 1, the r-Whitney-Eulerian polynomials satisfy the recurrence Proof.From recurrence (6) we get The log-concavity property of the Eulerian numbers can be proved by means of the real zero property of the Eulerian polynomials A n (x) (cf.[2]).A sequence {a 0 , a 1 , . . ., a n } of the coefficients of a polynomial f (x) = n k=0 a k x k of degree n with only real zeros is called the Pólya frequency sequence (PF).It is well know that if a sequence is PF then it is log-concave (cf.[2]).We are going to prove that the sequence A m,r (n, k) is a PF-sequence.To reach this aim, we first prove the following general lemma.Lemma 9. Let (T n (x)) n be a sequence of functions for n 0 defined by for some sequence of functions (p n (x)) n , (q n (x)) n , then where we define for some suitable real number α r n (x) = q n (x) u n (x) and u n (x) = e x α pn(t) qn(t) dt .
the electronic journal of combinatorics 25(1) (2018), #P1.48 Proof.Observe that Then Theorem 10.For n 1, the r-Whitney-Eulerian polynomials A n,m,r (x) have only nonpositive real roots if m r 0. Therefore (A m,r (n, k)) k is a PF-sequence.
Proof.The case in which m = r is clear because A m,m (n, k) = m n A(n, n − k − 1).Let us assume that m > r, this implies 0 < 1 − r m .Using our previous lemma and identity (8) we have that We now proceed by using induction over n.For n = 1 we get Proof.If we assume that n is prime, then the implication follows from Theorem 14.For the converse observe that r) n,k |.We are going to prove that the numbers b (m,r) n,k satisfy the same recurrence that A m,r (n, k) with the same initial values.Indeed, note that any (m, r)-coloured signed permutation of [n] with k descents can be obtained from a (m, r)coloured signed permutation π of [n − 1] with k or k − 1 descents by inserting the entries n or −n into π .

A 1 ,Theorem 11 .Theorem 16 .
m,r (x) = r + (m − r)x which have only one real root being x = − r m − r < 0. By the inductive hypothesis for n − 1 the term x r m (1 − x) −n A n−1,m,r (x) has n − 1 non-positive real roots plus the root in x = 0.So by Rolle's Theorem the derivative of this term must have exactly n − 1 non-positive real roots and by Equation (9) the polynomial A n,m,r (x) must have n − 1 non-positive real roots.Since complex roots appear in conjugate pairs the only choice for the last root of A n,m,r (x) is to be real and non positive since the polynomial A n,m,r (x) has positive coefficients.Therefore we have the following theorem.If 0 r m, the r-Whitney Eulerian sequence (A m,r (n, k)) k is log-concave and therefore unimodal.In particular, if m = r = 1 we have the following congruence for the Eulerian numbers.Corollary 15 ([12], Theorem 1).If p is a prime number and 1, 1 k p − 1, then A(p − 1, k) ≡ 1 (mod p), if p |k; 0 (mod p), otherwise.Let n be an integer such that n does not divide the integers (k + 2)m − r, (k + 1)m − r and m.Then n is prime if and only if A m,r (n − 1, k) ≡ 1 (mod n), for 1 k + 1 n − 1 and m n−1 ≡ [(k + 2)m − r] n−1 ≡ [(k + 1)m − r] n−1 ≡ 1 (mod n).