Related papers: Descent sets of cyclic permutations

We prove an identity conjectured by Adin and Roichman involving the descent set of $\lambda$-unimodal cyclic permutations. These permutations appear in formulas for characters of certain representations of the symmetric group. Such formulas…

Recently, Dokos et al. conjectured that for all $k, m\geq 1$, the patterns $ 12\ldots k(k+m+1)\ldots (k+2)(k+1) $ and $(m+1)(m+2)\ldots (k+m+1)m\ldots 21 $ are $maj$-Wilf-equivalent. In this paper, we confirm this conjecture for all $k\geq…

In 1916, MacMahon showed that permutations in $S_n$ with a fixed descent set $I$ are enumerated by a polynomial $d_I(n)$. Diaz-Lopez, Harris, Insko, Omar, and Sagan recently revived interest in this descent polynomial, and suggested the…

Visontai conjectured in 2013 that the joint distribution of ascent and distinct nonzero value numbers on the set of subexcedant sequences is the same as that of descent and inverse descent numbers on the set of permutations. This conjecture…

It is known that the number of permutations in the symmetric group $S_{2n}$ with cycles of odd lengths only is equal to the number of permutations with cycles of even lengths only. We prove a refinement of this equality, involving descent…

We present a bijection between permutation matrices and descending plane partitions without special parts, which respects the quadruple of statistics considered by Behrend, Di Francesco and Zinn--Justin. This bijection involves the…

Let S_n(321) (respectively, S_n(132)) denote the set of all permutations of {1,2,...,n} that avoid the pattern 321 (respectively, the pattern 132). Elizalde and Pak gave a bijection Theta from S_n(321) to S_n(132) that preserves the numbers…

We consider the two permutation statistics which count the distinct pairs obtained from the last two terms of occurrences of patterns t_1...t_{m-2}m(m-1) and t_1...t_{m-2}(m-1)m in a permutation, respectively. By a simple involution in…

Centrosymmetric involutions in the symmetric group S_{2n} are permutations \pi such that \pi=\pi^{-1} and \pi(i)+\pi(2n+1-i)=2n+1 for all i, and they are in bijection with involutions of the hyperoctahedral group. We describe the…

A permutation is (1-23-4)-avoiding if it contains no four entries, increasing left to right, with the middle two adjacent in the permutation. Here we give a 2-variable recurrence for the number of such permutations, improving on the…

In this note, we prove some and conjecture other results regarding the distribution of descent top and descent bottom sets on some pattern-avoiding permutations. In particular, for 3-letter patterns, we show bijectively that the set of…

The descent set D(w) of a permutation w of 1,2,...,n is a standard and well-studied statistic. We introduce a new statistic, the connectivity set C(w), and show that it is a kind of dual object to D(w). The duality is stated in terms of the…

Set partitions avoiding $k$-crossing and $k$-nesting have been extensively studied from the aspects of both combinatorics and mathematical biology. By using the generating tree technique, the obstinate kernel method and Zeilberger's…

We study the combinatorial equivalence of separable elements in types $A$ and $B$. A bijection is constructed from the set of separable permutations in the symmetric group $S_{n+1}$ to the set of separable signed permutations in the…

We define a map between the set of permutations that avoid either the four patterns $3214,3241,4213,4231$ or $3124,3142,4123,4132$, and the set of Dyck prefixes. This map, when restricted to either of the two classes, turns out to be a…

A bijection between $(31245,32145,31254,32154)$-avoiding permutations and $(31425,32415,31524,32514)$-avoiding permutations is constructed, which preserves five classical set-valued statistics. Combining with two codings of permutations due…

We present a bijection between non-crossing partitions of the set $[2n+1]$ into $n+1$ blocks such that no block contains two consecutive integers, and the set of sequences $\{s_{i}\}_{1}^{n}$ such that $1 \leq s_{i} \leq i$, and if…

We give some results about a bijection associating each permutation with a subexcedant function. This function is related to a particular decomposition of the permutation as a product of transpositions and therefore it has been called…

We present a simple bijection between Baxter permutations of size $n$ and plane bipolar orientations with n edges. This bijection translates several classical parameters of permutations (number of ascents, right-to-left maxima,…

Given sets X and Y of positive integers and a permutation sigma = sigma_1, sigma_2, ..., sigma_n in S_n, an X,Y-descent of sigma is a descent pair sigma_i > sigma_{i+1} whose "top" sigma_i is in X and whose "bottom" sigma_{i+1} is in Y. We…