Related papers: A recursive bijective approach to counting permuta…
We prove that deciding whether a given input word contains as subsequence every possible permutation of integers $\{1,2,\ldots,n\}$ is coNP-complete. The coNP-completeness holds even when given the guarantee that the input word contains as…
We study the generating function for the number of even (or odd) permutations on n letters containing exactly $r\gs0$ occurrences of 132. It is shown that finding this function for a given r amounts to a routine check of all permutations in…
Any permutation polynomial is an $ n $-cycle permutation. When $n$ is a specific small positive integer, one can obtain efficient permutations, such as involutions, triple-cycle permutations and quadruple-cycle permutations. These…
In this paper, we study three applications of recursion to problems in coding and random permutations. First, we consider locally recoverable codes with partial locality and use recursion to estimate the minimum distance of such codes. Next…
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…
The Goulden-Jackson cluster method is a powerful tool for counting words by occurrences of prescribed subwords, and was adapted by Elizalde and Noy for counting permutations by occurrences of prescribed consecutive patterns. In this paper,…
A permutation $\pi$ is ballot if, for all $k$, the word $\pi_1\cdots \pi_k$ has at least as many ascents as it has descents. Let $b(n)$ denote the number of ballot permutations of order $n$, and let $p(n)$ denote the number of permutations…
Permutations that avoid given patterns have been studied in great depth for their connections to other fields of mathematics, computer science, and biology. From a combinatorial perspective, permutation patterns have served as a unifying…
Each positive increasing integer sequence $\{a_n\}_{n\geq 0}$ can serve as a numeration system to represent each non-negative integer by means of suitable coefficient strings. We analyse the case of $k$-generalized Fibonacci sequences…
The generating polynomial of permutations of size $n$, counted by the number of alternating runs, has a root at $-1$ of multiplicity $\lfloor (n-2)/2 \rfloor$ for all $n \ge 2$. This result can be derived by combining the David--Barton…
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.
Babson and Steingr\`imsson introduced generalized permutation patterns that allow the requirement that two adjacent letters in a pattern must be adjacent in the permutation. Subsequently, Claesson presented a complete solution for the…
Canon permutations are permutations of the multiset having $k$ copies of each integer between $1$ and $n$, with the property that the subsequences obtained by taking the $j$th copy of each entry, for each fixed $j$, are all the same. For…
We prove that the total number $S_{n,132}(q)$ of copies of the pattern $q$ in all 132-avoiding permutations of length $n$ is the same for $q=231$, $q=312$, or $q=213$. We provide a combinatorial proof for this unexpected threefold symmetry.…
Baryshnikov and Romik derived the combinatorial identities for the numbers of the $m$-strip tableaux. This generalized the classical Andr\'e's theorem for the number of up-down permutations. They asked for a bijective proof for the…
We find generating functions the number of strings (words) containing a specified number of occurrences of certain types of order-isomorphic classes of substrings called subword patterns. In particular, we find generating functions for the…
Detecting and counting copies of permutation patterns are fundamental algorithmic problems, with applications in the analysis of rankings, nonparametric statistics, and property testing tasks such as independence and quasirandomness…
We prove an existing conjecture that the sequence defined recursively by $a_1=1, a_2=2, a_n=4a_{n-1}-2a_{n-2}$ counts the number of length-$n$ permutations avoiding the four generalized permutation patterns 1-32-4, 1-42-3, 2-31-4, and…
Permutation $\sigma$ appears in permutation $\pi$ if there exists a subsequence of $\pi$ that is order-isomorphic to $\sigma$. The natural question is to check if $\sigma$ appears in $\pi$, and if so count the number of occurrences. We know…
Given a permutation w, we show that the number of repeated letters in a reduced decomposition of w is always less than or equal to the number of 321- and 3412-patterns appearing in w. Moreover, we prove bijectively that the two quantities…