Consider S_n, the symmetric group on n letters, and let maj pi denote the major index of a permutation pi in S_n. Given positive integers k,l and nonnegative integers i,j, define m_n^{k,l}(i,j) := number of pi in S_n such that maj pi = i (mod k) and maj pi^{-1} = j (mod l). We prove bijectively that if k,l are relatively prime and at most n then m_n^{k,l}(i,j) = n!/(kl) which, surprisingly, does not depend on i and j. Equivalently, if m_n^{k,l}(i,j) is interpreted as the (i,j)-entry of a matrix m_n^{k,l}, then this is a constant matrix under the stated conditions. This bijection is extended to show the more general result that for d at least 1 and k,l relatively prime, the matrix m_n^{kd,ld} admits a block decompostion where each block is the matrix m_n^{d,d}/(kl). We also give an explicit formula for m_n^{n,n} and show that if p is prime then m_{np}^{p,p} has a simple block decomposition. To prove these results, we use the representation theory of the symmetric group and certain restricted shuffles.
Heaps of rhombic dodecahedra, catalan congruences on alternating sign matrices, and bases of the Temperley-Lieb algebra
LISN (UMR CNRS 9015), Université Paris Saclay, INRIA, CNRS（巴黎萨克雷大学）；Universitat de Barcelona & Centre de Recerca Matemàtica（巴塞罗那大学）；LIGM, Université Gustave Eiffel（巴黎东马恩拉瓦莱大学）
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).
This paper deals with lattice congruences of the weak order on the symmetric group, and initiates the investigation of the cover graphs of the corresponding lattice quotients. These graphs also arise as the skeleta of the so-called quotientopes, a family of polytopes recently introduced by Pilaud and Santos [Bull. Lond. Math. Soc., 51:406-420, 2019], which generalize permutahedra, associahedra, hypercubes and several other polytopes. We prove that all of these graphs have a Hamilton path, which can be computed by a simple greedy algorithm. This is an application of our framework for exhaustive…
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.
Given a prime $p\geq 5$, we reduce modulo p a convolution of order p-1 of powers of two weighted Bernoulli numbers with Bernoulli numbers in terms of harmonic numbers and generalized harmonic numbers. Our proof is based on studying the p-adic expansion of the number of permutations of Sym(p-2) with an even number of ascents, up to the modulus $p^2$.
We find a formula for the number of permutations of $[n]$ that have exactly $s$ runs up and down. The formula is at once terminating, asymptotic, and exact.
Let ${\mathcal C}_n$ be the set of all permutation cycles of length $n$ over $\{1,2,\ldots,n\}$. Let $${\mathfrak f}_n(q):=\sum_{σ\in{\mathcal C}_{n+1}}q^{{\mathrm maj}\,σ} $$ be a $q$-analogue of the factorial $n!$, where ${\mathrm maj}$ denotes the major index. We prove a $q$-analogue of Wilson's congruence $$ {\mathfrak f}_{n-1}(q)\equivμ(n)\pmod{Φ_n(q)}, $$ where $μ$ denotes the Möbius function and $Φ_n(q)$ is the $n$-th cyclotomic polynomial.
We extend George Andrew's general principle for counting generalized Frobenius partitions to include arrays with nonzero row difference and establish some congruences for these arrays.
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 $S_{2r}$.
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 $S_{2r}$.
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.
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.
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.
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…
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…
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.
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.
This paper presents a collection of experimental results regarding permutation pattern avoidance, focusing on cases where there are "many" patterns to be avoided.
For $G=G_{n, 1/2}$, the Erdős--Renyi random graph, let $X_n$ be the random variable representing the number of distinct partitions of $V(G)$ into sets $A_1, \ldots, A_q$ so that the degree of each vertex in $G[A_i]$ is divisible by $q$ for all $i\in[q]$. We prove that if $q\geq 3$ is odd then $X_n\xrightarrow{d}{\mathrm{Po}(1/q!)}$, and if $q \geq 4$ is even then $X_n\xrightarrow{d}{\mathrm{Po}(2^q/q!)}$. More generally, we show that the distribution is still asymptotically Poisson when we require all degrees in $G[A_i]$ to be congruent to $x_i$ modulo $q$ for each $i\in[q]$, where the residue…
In his Ph.D. thesis, Ira Gessel proved a reciprocity formula for noncommutative symmetric functions which enables one to count words and permutations with restrictions on the lengths of their increasing runs. We generalize Gessel's theorem to allow for a much wider variety of restrictions on increasing run lengths, and use it to complete the enumeration of permutations with parity restrictions on peaks and valleys, and to give a systematic method for obtaining generating functions for permutation statistics that are expressible in terms of increasing runs. Our methods can also be used to obtai…
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.
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, …
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…
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.
The m-Tamari lattice of F. Bergeron is an analogue of the clasical Tamari order defined on objects counted by Fuss-Catalan numbers, such as m-Dyck paths or (m+1)-ary trees. On another hand, the Tamari order is related to the product in the Loday-Ronco Hopf algebra of planar binary trees. We introduce new combinatorial Hopf algebras based on (m+1)-ary trees, whose structure is described by the m-Tamari lattices. In the same way as planar binary trees can be interpreted as sylvester classes of permutations, we obtain (m+1)-ary trees as sylvester classes of what we call m-permutations. These obje…
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…
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…
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…
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…
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.
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…
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 …
Given a set $Π$ of permutation patterns of length at most $k$, we present an algorithm for building $S_{\le n}(Π)$, the set of permutations of length at most $n$ avoiding the patterns in $Π$, in time $O(|S_{\le n - 1}(Π)| \cdot k + |S_{n}(Π)|)$. Additionally, we present an $O(n!k)$-time algorithm for counting the number of copies of patterns from $Π$ in each permutation in $S_n$. Surprisingly, when $|Π| = 1$, this runtime can be improved to $O(n!)$, spending only constant time per permutation. Whereas the previous best algorithms, based on generate-and-check, take exponential time per permutat…
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…
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…
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…
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…
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.
Given a surface with boundary and some points on its boundary, a polygon diagram is a way to connect those points as vertices of non-overlapping polygons on the surface. Such polygon diagrams represent non-crossing permutations on a surface with any genus and number of boundary components. If only bigons are allowed, then it becomes an arc diagram. The count of arc diagrams is known to have a rich structure. We show that the count of polygon diagrams exhibits the same interesting behaviours, in particular it is almost polynomial in the number of points on the boundary components, and the leadi…
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…
We show that the problem of counting collinear points in a permutation (previously considered by the author and J. Solymosi in "Collinear Points in Permutations", 2005) and the well-known finite plane Kakeya problem are intimately connected. Via counting arguments and by studying the hypergraph of collinear triples we show a new lower bound (5q/14 + O(1)) for the number of collinear triples of a permutation of GF(q) and a new lower bound (q(q + 1)/2 + 5q/14 + O(1)) on the size of the smallest Besicovitch set in GF(q)^2. Several interesting questions about the structure of the collinear triple …
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.
