Two identities of derangements (original) (raw)
A short combinatorial proof of derangement identity
Elemente der Mathematik
The n-th rencontres number with the parameter r is the number of permutations having exactly r fixed points. In particular, a derangement is a permutation without any fixed point. We presents a short combinatorial proof for a weighted sum derangement identities.
Barycentric subdivisions and derangement polynomials for the even-signed permutation groups
2012
The derangement polynomial for the symmetric group enumerates derangements by the number of excedances. It can be interpreted as the local hhh-polynomial, in the sense of Stanley, of the barycentric subdivision of the simplex. Motivated by this interpretation, we define a derangement polynomial for the even-signed permutation group. The coefficients of this polynomial are nonnegative, symmetric and unimodal. We show that they enumerate derangements in the even-signed permutation group according to a notion of excedance, which is analogous to the one introduced by Brenti for signed permutations. We also give an explicit formula for the corresponding exponential generating function.
A symmetric unimodal decomposition of the derangement polynomial of type BBB
The derangement polynomial d n (x) for the symmetric group enumerates derangements by the number of excedances. The derangement polynomial d B n (x) for the hyperoctahedral group is a natural type B analogue. A new combinatorial formula for this polynomial is given in this paper. This formula implies that d B n (x) decomposes as a sum of two nonnegative, symmetric and unimodal polynomials whose centers of symmetry differ by a half and thus provides a new transparent proof of its unimodality. A geometric interpretation, analogous to Stanley's interpretation of d n (x) as the local h-polynomial of the barycentric subdivision of the simplex, is given to one of the summands of this decomposition. This interpretation leads to a unimodal decomposition of the Eulerian polynomial of type B whose summands can be expressed in terms of the Eulerian polynomial of type A. The various decomposing polynomials introduced here are also studied in terms of recurrences, generating functions, combinatorial interpretations, expansions and real-rootedness.
2018
The classical derangement numbers count fixed point-free permutations. In this paper we study the enumeration problem of generalized derangements, when some of the elements are restricted to be in distinct cycles in the cycle decomposition. We find exact formula, combinatorial relations for these numbers as well as analytic and asymptotic description. Moreover, we study deeper number theoretical properties, like modularity, p-adic valuations, and diophantine problems.
Journal of Integer Sequences, 2003
In this paper we introduce some formulas for the number of derangements. Then we define the derangement function and use the software package MAPLE to obtain some integrals related to the incomplete gamma function and also to some hypergeometric summations.
A note on degenerate derangement polynomials and numbers
AIMS Mathematics, 2021
In this paper, we study the degenerate derangement polynomials and numbers, investigate some properties of those polynomials and numbers and explore their connections with the degenerate gamma distributions. In more detail, we derive their explicit expressions, recurrence relations and some identities involving the degenerate derangement polynomials and numbers and other special polynomials and numbers, which include the fully degenerate Bell polynomials, the degenerate Fubini polynomials and the degenerate Stirling numbers of both kinds. We also show that those polynomials and numbers are connected with the moments of some variants of the degenerate gamma distributions.
Counting derangements, involutions and unimodal elements in the wreath product C r ≀ S n
Israel Journal of Mathematics, 2010
We count derangements, involutions and unimodal elements in the wreath product Cr Sn by the numbers of excedances, fixed points and 2-cycles. Properties of the generating functions, including combinatorial formulas, recurrence relations and real-rootedness are studied. The results obtained specialize to those on the symmetric group Sn and on the hyperoctahedral group Bn when r = 1, 2, respectively.
On the rrr-Derangements of type B
2021
Consider two sets of n symbols [n] = {1, 2, . . . , n} and [n] = {1, 2, . . . , n}, with i 6= j, for any i, j. Define Xn = [n] ∪ [n]. The symbol j is called the colored version of the symbol j. Naturally there are (2n)! permutations of Xn. Some of these permutations respect the sign, that is, satisfy σ(j) = σ(j). These are called signed permutations or permutations of type B. General information about them and their relations to Coxeter groups appears in Section 8.1 of [5]. The number of signed permutations on [n] is 2nn!, since each one of them is formed by a permutation of [n] and a choice of sign. An example is
On two conjectures regarding generalized sequence of derangements
arXiv: Number Theory, 2020
The second author studied arithmetic properties of a class of sequences that generalize the sequence of derangements. The aim of the following paper is to disprove two conjectures stated in \cite{miska}. The first conjecture regards the set of prime divisors of their terms. The latter one is devoted to the order of magnitude of considered sequences.