Counting permutations by their runs up and down

Alternating runs of permutations and the central factorial numbers

Let R(n,k) be the number of permutations of $\{1,2,\ldots,n\}$ with k alternating runs. In this paper, we establish the relationships between R(n,k) and the central factorial numbers of even indices as well as the number of signed permutations with a given number of alternating runs and the central factorial numbers of odd indices. The explicit formulas of the peak and left peak polynomials for permutations and the derivative polynomials of the tangent and secant functions are also established.

2-stack sortable permutations with a given number of runs

Using earlier results we prove a formula for the number $W_{(n,k)}$ of 2-stack sortable permutations of length $n$ with $k$ runs, or in other words, $k-1$ descents. This formula will yield the suprising fact that there are as many 2-stack sortable permutations with $k-1$ descents as with $k-1$ ascents. We also prove that $W_{(n,k)}$ is unimodal in $k$, for any fixed $n$.

Counting stabilized-interval-free permutations

A stabilized-interval-free (SIF) permutation on [n]={1,2,...,n} is one that does not stabilize any proper subinterval of [n]. By presenting a decomposition of an arbitrary permutation into a list of SIF permutations, we show that the generating function A(x) for SIF permutations satisfies the defining property: [x^(n-1)] A(x)^n = n! . We also give an efficient recurrence for counting SIF permutations.

Counting occurences of 132 in a permutation

We study the generating function for the number of permutations on n letters containing exactly $r\gs0$ occurences of 132. It is shown that finding this function for a given r amounts to a routine check of all permutations in $S_{2r}$.

Counting occurrences of 132 in an even permutation

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 $S_{2r}$.

Counting pop-stacked permutations in polynomial time

Permutations in the image of the pop-stack operator are said to be pop-stacked. We give a polynomial-time algorithm to count pop-stacked permutations up to a fixed length and we use it to compute the first 1000 terms of the corresponding counting sequence. Only the first 16 terms had previously been computed. With the 1000 terms we prove some negative results concerning the nature of the generating function for pop-stacked permutations. We also predict the asymptotic behavior of the counting sequence using differential approximation.

Counting Signed Vexillary Permutations

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 generating tree developed by West. Many further directions are mentioned in the end.

计数无不动点的Cayley排列

Counting fixed-point-free Cayley permutations

通过双种类方法得到Cayley排列的函数有向图的微分方程，推导出无不动点Cayley排列的显式公式，并证明其比例趋于1/e，与排列和自映射一致。

Recurrence relations in counting the pattern 13-2 in flattened permutations

We prove that the generating function for the number of flattened permutations having a given number of occurrences of the pattern 13-2 is rational, by using the recurrence relations and the kernel method.

Counting crucial permutations with respect to monotone patterns

Recently, Avgustinovich, Kitaev, and Taranenko defined five types of $(k, \ell)-$crucial permutations, which are maximal permutations that do not contain an increasing subsequence of length $k$ or a decreasing subsequence of length $\ell$. Further, Avgustinovich, Kitaev, and Taranenko began the enumeration of the $(k, \ell)-$crucial permutations of the minimal length and the next minimal length and the $(k, 3)-$crucial permutations of all lengths for each of the five types of $(k,\ell)-$crucial permutations. In this paper, we complete the enumeration that Avgustinovich, Kitaev, and Taranenko b…

Counting genus one partitions and permutations

We prove the conjecture by M. Yip stating that counting genus one partitions by the number of their elements and parts yields, up to a shift of indices, the same array of numbers as counting genus one rooted hypermonopoles. Our proof involves representing each genus one permutation by a four-colored noncrossing partition. This representation may be selected in a unique way for permutations containing no trivial cycles. The conclusion follows from a general generating function formula that holds for any class of permutations that is closed under the removal and reinsertion of trivial cycles. Ou…

A new refinement of Euler numbers on counting alternating permutations

At a crossroads of calculus and combinatorics, the generating function of secant and tangent numbers (Euler numbers) provides enumeration of alternating permutations. In this article, we present a new refinement of Euler numbers to answer the combinatorial question on some particular relation of Euler numbers proved by Heneghan-Petersen, Power series for up-down min-max permutations, College Math. Journal, Vol. 45, No. 2 (2014), 83-91.

布尔排列的 Runs 与 RSK 表

Runs and RSK tableaux of boolean permutations

本文为布尔排列定义规范约化词，据此直接读出其 RSK 表，描述并枚举对应表，并将 run 统计量对 RSK 形状的影响推广到任意排列。

Counting Fixed-Length Permutation Patterns

We consider the problem of packing fixed-length patterns into a permutation, and develop a connection between the number of large patterns and the number of bonds in a permutation. Improving upon a result of Kaplansky and Wolfowitz, we obtain exact values for the expectation and variance for the number of large patterns in a random permutation. Finally, we are able to generalize the idea of bonds to obtain results on fixed-length patterns of any size, and present a construction that maximizes the number of distinct large patterns.

Counting Permutations that Avoid Many Patterns

This paper presents a collection of experimental results regarding permutation pattern avoidance, focusing on cases where there are "many" patterns to be avoided.

A Grammatical Calculus for Peaks and Runs of Permutations

We develop a nonstandard approach to exploring polynomials associated with peaks and runs of permutations. With the aid of a context-free grammar, or a set of substitution rules, one can perform a symbolic calculus, and the computation often becomes rather simple. From a grammar it follows at once a system of ordinary differential equations for the generating functions. Utilizing a certain constant property, it is even possible to deduce a single equation for each generating function. To bring the grammar to a combinatorial setting, we find a labeling scheme for up-down runs of a permutation, …

Bijective counting of involutive Baxter permutations

We enumerate bijectively the family of involutive Baxter permutations according to various parameters; in particular we obtain an elementary proof that the number of involutive Baxter permutations of size $2n$ with no fixed points is $\frac{3\cdot 2^{n-1}}{(n+1)(n+2)}\binom{2n}{n}$, a formula originally discovered by M. Bousquet-Mélou using generating functions. The same coefficient also enumerates planar maps with $n$ edges, endowed with an acyclic orientation having a unique source, and such that the source and sinks are all incident to the outer face.

Counting King Permutations on the Cylinder

We call a permutation $σ=[σ_1,\dots,σ_n] \in S_n$ a {\em cylindrical king permutation} if $ |σ_i-σ_{i+1}|>1$ for each $1\leq i \leq n-1$ and $|σ_1-σ_n|>1$. We present some results regarding the distribution of the cylindrical king permutations, including some interesting recursions. We also calculate their asymptotic proportion in the set of the 'king permutations', i.e. the ones which satisfy only the first of the two conditions above. With this aim we define a new parameter on permutations, namely, the number of {\em cyclic bonds} which is a modification of the number of bonds. In addition, …

通过Hetyei-Reiner树按交替运行计数排列

Counting permutations by alternating runs via Hetyei-Reiner trees

本文通过Hetyei-Reiner树作用，提出新的方法解释Bóna商多项式，并扩展到B和D类型排列，获得相关恒等式组合证明。

On counting permutations by pairs of congruence classes of major index

For a fixed positive integer n, let S_n denote the symmetric group of n! permutations on n symbols, and let maj(sigma) denote the major index of a permutation sigma. For positive integers k<m not greater than n and non-negative integers i and j, we give enumerative formulas for the cardinality of the set of permutations sigma in S_n with maj(sigma) congruent to i mod k and maj(sigma^(-1)) congruent to j mod m. When m divides n-1 and k divides n, we show that for all i,j, this cardinality equals (n!)/(km).

Counting permutations by alternating descents

We find the exponential generating function for permutations with all valleys even and all peaks odd, and use it to determine the asymptotics for its coefficients, answering a question posed by Liviu Nicolaescu. The generating function can be expressed as the reciprocal of a sum involving Euler numbers. We give two proofs of the formula. The first uses a system of differential equations. The second proof derives the generating function directly from general permutation enumeration techniques, using noncommutative symmetric functions. The generating function is an "alternating" analogue of Davi…

计数上-上或下-下排列

Counting up-up-or-down-down permutations

本文回答了Donald Knuth的问题，通过分析奇数长度上-上或下-下排列的最后一个元素，求出了其双变量指数生成函数，刻画了峰和谷均位于奇数位置的排列结构。

减少多重集拟斯特林排列中的降序游程

Decreasing Runs in Quasi-Stirling Permutations of Multisets

本文证明多重集拟斯特林排列的降序游程具有分布不变性，构造保留降序游程的多重度重分配双射，结合经典游程定理导出相关统计量的递推关系与生成函数。

扁平化斯特林排列

Flattened Stirling Permutations

本文研究扁平化斯特林排列，建立其与类型B集合划分的双射，从而给出Dowling数计数的新证明，并进一步分析run的数量分布及相关枚举问题。

计数其乘积为具有受限循环长度的排列的循环对

Counting pairs of cycles whose product is a permutation with restricted cycle lengths

本文研究乘积循环长度受限时 $N$-循环对的计数，给出精确与渐近公式，并据此证明与恒等排列块转置距离至少为 $(n+1)/2$ 的排列数量具有较高下界。

计数对称群$S_{2n}$与$S_{2n+1}$中的置换

Counting Permutations in $S_{2n}$ and $S_{2n+1}$

本文针对Stanley提出的对称群中完全平方置换计数猜想，给出组合证明，并揭示两类对称群中含偶数个偶循环的置换的三类对应分类关系。

避免$\boldsymbol{\u0332{32}}1$模式的排列的逆降枚举器

Inversion-descent enumerators of $\underline{32}1$-avoiding permutations

该研究针对避免$\boldsymbol{321}$ vincular模式的排列，推导了涉及逆序、降序的计数多项式递推关系与$q$-指数生成函数，并从受限增长单词角度提供了新解释。

避免两个平坦部分有序模式的排列计数

Counting permutations avoiding two flat partially ordered patterns

研究同时避免两个平坦部分有序模式的排列计数问题，通过建立与k-斐波那契数的联系和双射方法，推导生成函数并扩展了单个平坦POPs的计数结果。

计数具有受限前缀和后缀的交错排列

Counting alternating permutations with restricted prefix and suffix

该研究借助 Hasse 图与生成函数，枚举长度3、4的前缀或后缀满足模式及数值约束的交错排列，并处理前后缀同时受限情形。

Counting (3+1) - Avoiding permutations

A poset is {\it $(\3+\1)$-free} if it contains no induced subposet isomorphic to the disjoint union of a 3-element chain and a 1-element chain. These posets are of interest because of their connection with interval orders and their appearance in the $(\3+\1)$-free Conjecture of Stanley and Stembridge. The dimension 2 posets $P$ are exactly the ones which have an associated permutation $π$ where $i\prec j$ in $P$ if and only if $i<j$ as integers and $i$ comes before $j$ in the one-line notation of $π$. So we say that a permutation $π$ is {\it $(\3+\1)$-free} or {\it $(\3+\1)$-avoiding} if its p…

A recursive bijective approach to counting permutations containing 3-letter patterns

We present a method, illustrated by several examples, to find explicit counts of permutations containing a given multiset of three letter patterns. The method is recursive, depending on bijections to reduce to the case of a smaller multiset, and involves a consideration of separate cases according to how the patterns overlap. Specifically, we use the method (i) to provide combinatorial proofs of Bona's formula {2n-3}choose{n-3} for the number of n-permutations containing one 132 pattern and Noonan's formula 3/n {2n}choose{n+3} for one 123 pattern, (ii) to express the number of n-permutations c…

Counting Standard Young Tableaux With Restricted Runs

The number of Young Tableaux whose shape is a k by n rectangle is famously (nk)! 0! ... (k-1)!/((n+k-1)!(n+k-2)!... n!) implying that for each specific k, that sequence satisfies a linear recurrence equation with polynomial coefficients of the first order. But what about counting Young tableaux where certain "run lengths" are forbidden? Then things seem to get much more complicated. We conclude with four conjectures and pledge donations to the OEIS in honor of the first provers.

Counting subwords in flattened permutations

In this paper, we consider the number of occurrences of descents, ascents, 123-subwords, 321-subwords, peaks and valleys in flattened permutations, which were recently introduced by Callan in his study of finite set partitions. For descents and ascents, we make use of the kernel method and obtain an explicit formula (in terms of Eulerian polynomials) for the distribution on $\mathcal{S}_n$ in the flattened sense. For the other four patterns in question, we develop a unified approach to obtain explicit formulas for the comparable distributions. We find that the formulas so obtained for 123- and…

Counting occurrences of a pattern of type (1,2) or (2,1) in permutations

Babson and Steingr\'ımsson introduced generalized permutation patterns that allow the requirement that two adjacent letters in a pattern must be adjacent in the permutation. Claesson presented a complete solution for the number of permutations avoiding any single pattern of type $(1,2)$ or $(2,1)$. For eight of these twelve patterns the answer is given by the Bell numbers. For the remaining four the answer is given by the Catalan numbers. With respect to being equidistributed there are three different classes of patterns of type $(1,2)$ or $(2,1)$. We present a recursion for the number of perm…

Consecutive patterns in circular permutations

In their study of cyclic pattern containment, Domagalski et al. conjecture differential equations for the generating functions of circular permutations avoiding consecutive patterns of length 3. In this note, we prove and significantly generalize these conjectures. We show that, for every consecutive pattern $σ$ beginning with 1, the bivariate generating function counting occurrences of $σ$ in circular permutations can be obtained from the generating function counting occurrences of $σ$ in (linear) permutations. This includes all the patterns for which the latter generating function is known.

The complexity of counting poset and permutation patterns

We introduce a notion of pattern occurrence that generalizes both classical permutation patterns as well as poset containment. Many questions about pattern statistics and avoidance generalize naturally to this setting, and we focus on functional complexity problems -- particularly those that arise by constraining the order dimensions of the pattern and text posets. We show that counting the number of induced, injective occurrences among dimension 2 posets is #P-hard; enumerating the linear extensions that occur in realizers of dimension 2 posets can be done in polynomial time, while for uncons…

On the number of permutations with bounded run lengths

In this work we obtain recurrent formulae for the number of permutations with either increasing or monotonic (i.e., both increasing and decreasing) runs of bounded length. Our formulae allow one to efficiently compute the number of such permutations. In particular, we use the formulae to find and correct a few miscalculations in the classic 1966 book by David, Kendall, and Barton. We further use our formulae to derive differential equations for the corresponding exponential generating functions. In the case of increasing runs, we solve these equations and obtain closed-form expressions for the…

Enumerating Permutations by their Run Structure

Motivated by a problem in quantum field theory, we study the up and down structure of circular and linear permutations. In particular, we count the length of the (alternating) runs of permutations by representing them as monomials and find that they can always be decomposed into so-called `atomic' permutations introduced in this work. This decomposition allows us to enumerate the (circular) permutations of a subset of the natural numbers by the length of their runs. Furthermore, we rederive, in an elementary way and using the methods developed here, a result due to Kitaev on the enumeration of…

Signed Alternating-runs enumeration in Classical Weyl Groups

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 that when enumerated with sign taken into account, one gets a {\it neat formula}. As a consequence, we get a near refinement of a result of Wilf on the exponent of $(1+t)$ when it divides the alternating-runs polynomial in the alternating group $\mathcal{A}_n$. Other applications include a moment-type identity and enumeration of alternating permutations in $\…

Subword counting and the incidence algebra

The Pascal matrix, $P$, is an upper diagonal matrix whose entries are the binomial coefficients. In 1993 Call and Velleman demonstrated that it satisfies the beautiful relation $P=\exp(H)$ in which $H$ has the numbers 1, 2, 3, etc. on its superdiagonal and zeros elsewhere. We generalize this identity to the incidence algebras $I(A^*)$ and $I(\mathcal{S})$ of functions on words and permutations, respectively. In $I(A^*)$ the entries of $P$ and $H$ count subwords; in $I(\mathcal{S})$ they count permutation patterns. Inspired by vincular permutation patterns we define what it means for a subword …

Counting descent pairs with prescribed colors in the colored permutation groups

We define new statistics, (c, d)-descents, on the colored permutation groups Z_r \wr S_n and compute the distribution of these statistics on the elements in these groups. We use some combinatorial approaches, recurrences, and generating functions manipulations to obtain our results.

Counting Permutations Modulo Pattern-Replacement Equivalences for Three-Letter Patterns

We study a family of equivalence relations on $S_n$, the group of permutations on $n$ letters, created in a manner similar to that of the Knuth relation and the forgotten relation. For our purposes, two permutations are in the same equivalence class if one can be reached from the other through a series of pattern-replacements using patterns whose order permutations are in the same part of a predetermined partition of $S_c$. When the partition is of $S_3$ and has one nontrivial part and that part is of size greater than two, we provide formulas for the number of classes created in each previous…

Counting Permutations Where The Difference Between Entries Located $r$ Places Apart Can never be $s$ (For any given positive integers $r$ and $s$)

Given positive integers $r$ and $s$, we use inclusion-exclusion, weighted-counting of tilings, and dynamical programming, in order to enumerate, semi-efficiently, the classes of permutations mentioned in the title. In the process we revisit beautiful previous work of Enrique Navarrete, Robert Tauraso, David Robbins (to whose memory this article is dedicated), and John Riordan. We also present two new proofs of John Riordan's recurrence (from 1965) for the sequence enumerating permutations without rising and falling successions (the $r = 1$, $s = 1$ case of the title in the sense of absolute va…

Signed Countings of types B and D permutations and $t,q$-Euler Numbers

It is a classical result that the parity-balance of the number of weak excedances of all permutations (derangements, respectively) of length $n$ is the Euler number $E_n$, alternating in sign, if $n$ is odd (even, respectively). Josuat-Vergès obtained a $q$-analog of the results respecting the number of crossings of a permutation. One of the goals in this paper is to extend the results to the permutations (derangements, respectively) of types B and D, on the basis of the joint distribution in statistics excedances, crossings and the number of negative entries obtained by Corteel, Josuat-Vergès…

Counting Parking Sequences and Parking Assortments Through Permutations

Parking sequences (a generalization of parking functions) are defined by specifying car lengths and requiring that a car attempts to park in the first available spot after its preference. If it does not fit there, then a collision occurs and the car fails to park. In contrast, parking assortments generalize parking sequences (and parking functions) by allowing cars (also of assorted lengths) to seek forward from their preference to identify a set of contiguous unoccupied spots in which they fit. We consider both parking sequences and parking assortments and establish that the number of prefere…

Applying the Cluster Method to Count Occurrences of Generalized Permutation Patterns

We apply ideas from the cluster method to q-count the permutations of a multiset according to the number of occurrences of certain generalized patterns, as defined by Babson and Steingrimsson. In particular, we consider those patterns with three letters and one internal dash, as well as permutation statistics composed of counting the number of occurrences of multisets of such patterns. Counting is done via recurrences which simplify in the case of permutations. A collection of Maple procedures implementing these recurrences accompanies the article.

A new record for $1324$-avoiding permutations

Refining an existing counting argument, we provide an improved upper bound for the number of 1324-avoiding permutations of a given length.

通过生成树与核方法计数无长单调子序列的排列

Counting permutations with no long monotone subsequence via generating trees and the kernel method

本文通过生成树和核方法，重新证明并推广了Gessel关于无长单调子序列排列的生成函数行列式公式，并扩展到对合情形，同时跟踪固定点数。

按大降标计数模式避免排列

Counting pattern-avoiding permutations by big descents

研究避免长度为3模式集合的排列中大降标统计量的分布，将大小为1和2的模式集分类为等价类并推导分布公式，方法涉及生成函数与双射。

Merging-Free Partitions and Run-Sorted Permutations

In this paper, we study merging-free partitions with their canonical forms and run-sorted permutations. We give a combinatorial proof of the conjecture made by Nabawanda et al. We describe the distribution of the statistics of runs and right-to-left minima over the set of run-sorted permutations and we give the exponential generating function for their joint distribution. We show the number of right-to-left minima is given by the shifted distribution of the Stirling number of the second kind. We also prove that the non-crossing merging-free partitions are enumerated by powers of 2. We use one …

Signed Countings of Type B and D Permutations and $t,q$-Euler numbers

A classical result states that the parity balance of the number of excedances of all permutations (derangements, respectively) of length $n$ is the Euler number. In 2010, Josuat-Vergès gives a $q$-analogue with $q$ representing the number of crossings. We extend this result to the permutations (derangements, respectively) of type B and D. It turns out that the signed countings are related to the derivative polynomials of $\tan$ and $\sec$. Springer numbers defined by Springer can be regarded as an analogue of Euler numbers defined on every Coxeter group. In 1992 Arnol'd showed that the Springe…

On the least exponential growth admitting uncountably many closed permutation classes

We show that the least exponential growth of counting functions which admits uncountably many closed permutation classes lies between 2^n and (2.33529...)^n.

随机非均匀排列中的记录计数

Counting records in a random, non-uniform, permutation

针对非均匀随机排列的最大期望记录数阶的界定问题，本文在特定参数条件下推导出其精确阶为$(1-p)\sqrt{n/p}$，完善了此前的上下界结果。

第二类欧拉数的恒等式：通过格点计数

An identity for second Eulerian numbers via lattice-point counting

利用 Ehrhart 理论，给出 Bernoulli 数与第二类欧拉数之间两个恒等式的简单概念性证明。