Related papers: Counting alternating permutations with restricted …
We present a bijection between 321- and 132-avoiding permutations that preserves the number of fixed points and the number of excedances. This gives a simple combinatorial proof of recent results of Robertson, Saracino and Zeilberger, and…
The Euler number $E_n$ (resp. Entringer number $E_{n,k}$) enumerates the alternating (down-up) permutations of $\{1,\dots,n\}$ (resp. starting with $k$). The Springer number $S_n$ (resp. Arnold number $S_{n,k}$) enumerates the type $B$…
Alternating sign trapezoids have recently been introduced as a generalisation of alternating sign triangles. Fischer established a threefold refined enumeration of alternating sign trapezoids and provided three statistics on column strict…
Considering commutator monomials of the non-commutative associative variables $X_1,\ldots,X_n$; we determine the maximal possible number of alternating associative monomials in their noncommutative polynomial expansions. This is achieved by…
Define $S_n^k(\alpha)$ to be the set of permutations of $\{1,2,...,n\}$ with exactly $k$ fixed points which avoid the pattern $\alpha \in S_m$. Let $s_n^k(\alpha)$ be the size of $S_n^k(\alpha)$. We investigate $S_n^0(\alpha)$ for all…
The sequence a_1,...,a_m is a common subsequence in the set of permutations S = {p_1,...,p_k} on [n] if it is a subsequence of p_i(1),...,p_i(n) and p_j(1),...,p_j(n) for some distinct p_i, p_j in S. Recently, Beame and Huynh-Ngoc (2008)…
We study the expansions of permutation statistics in the basis of functions counting occurrences of a fixed pattern in a permutation. We show the finiteness of these pattern expansions for a class of permutation statistics including the…
We initiate a general approach for the fast enumeration of permutations with a prescribed number of occurrences of `forbidden' patterns, that seems to indicate that the enumerating sequence is always P-recursive. We illustrate the method…
We study the total number of occurrences of several vincular (also called generalized) patterns and other statistics, such as the major index and the Denert statistic, on permutations avoiding a pattern of length 3, extending results of…
We enumerate and characterize some classes of alternating and reverse alternating involutions avoiding a single pattern of length three or four. If on one hand the case of patterns of length three is trivial, on the other hand, the length…
The extension of pattern avoidance from ordinary permutations to those on multisets gave birth to several interesting enumerative results. We study permutations on regular multisets, i.e., multisets in which each element occurs the same…
Given a set S of integers whose sum is zero, consider the problem of finding a permutation of these integers such that: (i) all prefix sums of the ordering are nonnegative, and (ii) the maximum value of a prefix sum is minimized. Kellerer…
The alternating-runs polynomial enumerates alternating runs in the symmetric group. There are three formulae for the number of permutations, $R_{n,k}$ in $\mathfrak{S}_n$ with $k$ alternating runs, but all of them are complicated. We show…
Prefix parsing asks whether an input prefix can be extended to a complete string generated by a given grammar. In the weighted setting, it also provides prefix probabilities, which are central to context-free language modeling,…
We show that the number of signed permutations avoiding 1234 equals the number of signed permutations avoiding 2143 (also called vexillary signed permutations), resolving a conjecture by Anderson and Fulton. The main tool that we use is the…
We consider the problem of counting the copies of a length-$k$ pattern $\sigma$ in a sequence $f \colon [n] \to \mathbb{R}$, where a copy is a subset of indices $i_1 < \ldots < i_k \in [n]$ such that $f(i_j) < f(i_\ell)$ if and only if…
Let $\sigma$ and $\tau$ be patterns of length three; that is $\sigma, \tau \in \{123,132,213,231,312,321\}$. In this paper, we enumerate the set of cyclic permutations in $\mathcal{S}_n$ that avoid $\sigma$ in their one-line notation and…
In the last decade a huge amount of articles has been published studying pattern avoidance on permutations. From the point of view of enumeration, typically one tries to count permutations avoiding certain patterns according to their…
We provide a simple injective proof that the number of 132-avoiding permutations with a unique longest increasing subsequence is at least as large as the number of 132-avoiding permutations without a unique longest increasing subsequence.
We call a permutation $\sigma=[\sigma_1,\dots,\sigma_n] \in S_n$ a {\em cylindrical king permutation} if $ |\sigma_i-\sigma_{i+1}|>1$ for each $1\leq i \leq n-1$ and $|\sigma_1-\sigma_n|>1$. We present some results regarding the…