In (Striker in Discret Math Theor Comput Sci 20, 2018), Striker generalized Cameron and Fon-Der-Flaass’s notion of a toggle group. In this paper, we begin the study of transitive generalized toggle groups that contain a cycle. We first show that if such a group has degree n and contains a transposition or a 3-cycle, then the group contains $$A_n$$ A n . Using the result about transpositions, we then prove that a transitive generalized toggle group that contains a short cycle must be primitive. Employing a result of Jones (Bull Aust Math Soc 89(1):159-165, 2014), which relies on the classification of the finite simple groups, we conclude that any transitive generalized toggle group of degree n that contains a cycle with at least 3 fixed points must also contain $$A_n$$ A n . Finally, we look at imprimitive generalized toggle groups containing a long cycle and show that they decompose into a direct product of primitive generalized toggle groups each containing a long cycle.

We prove a conjecture of Guo and Poznanović concerning chains in certain 01-fillings of moon polyominoes. A key ingredient of our proof is a correspondence between words w and pairs (W(w),M(w)) of increasing tableaux such that M(w) determines the lengths of the longest strictly increasing and strictly decreasing sequences in every subinterval of w. We define this correspondence by using Thomas and Yong’s K-infusion operator and then use it to obtain the bijections that prove the conjecture of Guo and Poznanović. In constructing our bijections we introduce new variants of the RSK correspondence and Knuth equivalence.

The following long-standing problem in combinatorics was first posed in 1993 by Gessel and Reutenauer. For which multisubsets $B$ of the symmetric group $\fS_n$ is the quasisymmetric function $$Q(B) = \sum_{\pi \in B}F_{\Des(\pi), n}$$ a symmetric function? Here $\Des(\pi)$ is the descent set of $\pi$ and $F_{\Des(\pi), n}$ is Gessel's fundamental basis for the vector space of quasisymmetric functions. The purpose of this paper is to provide a useful characterization of these multisets. Using this characterization we prove a conjecture of Elizalde and Roichman. Two other corollaries are also given. The first is a short new proof that conjugacy classes are symmetric sets, a well known result first proved by Gessel and Reutenauer. Our second corollary is a unified explanation that both left and right multiplication of symmetric multisets, by inverse $J$-classes, is symmetric. The case of right multiplication was first proved by Elizalde and Roichman.

Let S_n be the nth symmetric group. Given a set of permutations Pi we denote by S_n(Pi) the set of permutations in S_n which avoid Pi in the sense of pattern avoidance. Consider the generating function Q_n(Pi) = sum_pi F_{Des pi} where the sum is over all pi in S_n(Pi) and F_{Des pi} is the fundamental quasisymmetric function corresponding to the descent set of pi. Hamaker, Pawlowski, and Sagan introduced Q_n(Pi) and studied its properties, in particular, finding criteria for when this quasisymmetric function is symmetric or even Schur nonnegative for all n >= 0. The purpose of this paper is to continue their investigation answering some of their questions, proving one of their conjectures, as well as considering other natural questions about Q_n(Pi). In particular we look at Pi of small cardinality, superstandard hooks, partial shuffles, Knuth classes, and a stability property.

A partition $$\alpha $$ is said to contain another partition (or pattern) $$\mu $$ if the Ferrers board for $$\mu $$ is attainable from $$\alpha $$ under removal of rows and columns. We say $$\alpha $$ avoids $$\mu $$ if it does not contain $$\mu $$ . In this paper we count the number of partitions of n avoiding a fixed pattern $$\mu $$ , in terms of generating functions and their asymptotic growth rates. We find that the generating function for this count is rational whenever $$\mu $$ is (rook equivalent to) a partition in which any two part sizes differ by at least two. In doing so, we find a surprising connection to metacyclic p-groups. We further obtain asymptotics for the number of partitions of n avoiding a pattern $$\mu $$ . Using these asymptotics we conclude that the generating function for $$\mu $$ is not algebraic whenever $$\mu $$ is rook equivalent to a partition with distinct parts whose first two parts are positive and differ by 1.

Stankova and West proved in 2002 that the patterns 231 and 312 are shape-Wilf-equivalent. Their proof was nonbijective. We give a new characterization of 231 and 312 avoiding full rook placements and use this to give a simple bijection that demonstrates the shape-Wilf-equivalence.

We introduce a notion of {\em cyclic Schur-positivity} for sets of permutations, which naturally extends the classical notion of Schur-positivity, and it involves the existence of a bijection from permutations to standard Young tableaux that preserves the cyclic descent set. Cyclic Schur-positive sets of permutations are always Schur-positive, but the converse does not hold, as exemplified by inverse descent classes, Knuth classes and conjugacy classes. In this paper we show that certain classes of permutations invariant under either horizontal or vertical rotation are cyclic Schur-positive. The proof unveils a new equidistribution phenomenon of descent sets on permutations, provides affirmative solutions to conjectures from [9] and [2], and yields new examples of Schur-positive sets.

In Bloom and Saracino (2018) we introduced a new notion of Wilf equivalence of integer partitions and proved that rook equivalence implies Wilf equivalence. In the present paper we prove the converse and thereby establish a new criterion for rook equivalence. We also refine two of the standard criteria for rook equivalence and establish another new one involving what we call nested sequences of L’s.

The subjects of rook equivalence and Wilf equivalence have both attracted considerable attention over the last half-century. In this paper we introduce a new notion of Wilf equivalence for integer partitions, and, using this notion, we prove that rook equivalence implies Wilf equivalence. We also prove that if we refine the notions of rook and Wilf equivalence in a natural way, then these two notions coincide. In Bloom and Saracino (2017) we prove that Wilf equivalence implies rook equivalence.

Egge conjectured that permutations avoiding the set of patterns $\{2143,3142,\tau\}$, where $\tau\in\{246135,254613,263514,524361,546132\}$, are enumerated by the large Schr\"oder numbers. Consequently, $\{2143,3142,\tau\}$ with $\tau$ as above is Wilf-equivalent to the set of patterns $\{2413,3142\}$. Burstein and Pantone proved the case of $\tau=246135$. We prove the remaining four cases. As a byproduct of our proof, we also enumerate the case $\tau=4132$.

Let G be a group acting on a set X of combinatorial objects, with finite orbits, and consider a statistic xi : X -> C. Propp and Roby defined the triple (X, G, xi) to be homomesic if for any orbits O-1, O-2, the average value of the statistic xi is the same, that is1/vertical bar O-1 vertical bar Sigma(x is an element of O1)xi(x) = 1/vertical bar O-2 vertical bar(y is an element of O1)xi(y).In 2013 Propp and Roby conjectured the following instance of homomesy. Let SSYTk(m x n) denote the set of semistandard Young tableaux of shape mxn with entries bounded by k. Let S be any set of boxes in them x n rectangle fixed under 180 degrees rotation. For T is an element of SSYTk(m x n), define sigma(s)(T) to be the sum of the entries of T in the boxes of S. Let (P) be a cyclic group of order k where P acts on SSYTk(m x n) by promotion. Then (SSYTk(m x n), < P >, sigma(s)) is homomesic.We prove this conjecture, as well as a generalization to cominuscule posets. We also discuss analogous questions for tableaux with strictly increasing rows and columns under the K-promotion of Thomas and Yong, and prove limited results in that direction. (C) 2015 Elsevier B.V. All rights reserved.

Using bijections between pattern-avoiding permutations and certain full rook placements on Ferrers boards, we give short proofs of two enumerative results. The first is a simplified enumeration of the 3124, 1234avoiding permutations, obtained recently by Callan via a complicated decomposition. The second is a streamlined bijection between 1342-avoiding permutations and permutations which can be sorted by two increasing stacks in series, originally due to Atkinson, Murphy, and Ruskuc.

In 2000 Klazar introduced a new notion of pattern avoidance in the context of set partitions of $[n]=\{1,\ldots, n\}$. The purpose of the present paper is to undertake a study of the concept of Wilf-equivalence based on Klazar's notion. We determine all Wilf-equivalences for partitions with exactly two blocks, one of which is a singleton block, and we conjecture that, for $n\geq 4$, these are all the Wilf-equivalences except for those arising from complementation. If $\tau$ is a partition of $[k]$ and $\Pi_n(\tau)$ denotes the set of all partitions of $[n]$ that avoid $\tau$, we establish inequalities between $|\Pi_n(\tau_1)|$ and $|\Pi_n(\tau_2)|$ for several choices of $\tau_1$ and $\tau_2$, and we prove that if $\tau_2$ is the partition of $[k]$ with only one block, then $|\Pi_n(\tau_1)| k$ and all partitions $\tau_1$ of $[k]$ with exactly two blocks. We conjecture that this result holds for all partitions $\tau_1$ of $[k]$. Finally, we enumerate $\Pi_n(\tau)$ for all partitions $\tau$ of $[4]$.

In their paper [6], Dokos et al. conjecture that the major index statistic is equidistributed among 1423-avoiding, 2413-avoiding, and 3214-avoiding permutations. In this paper we confirm this conjecture by constructing two major index preserving bijections, Θ:Sn(1423)→Sn(2413) and Ω:Sn(3214)→Sn(2413). In fact, we show that Θ (respectively, Ω) preserves numerous other statistics including the descent set, right-to-left maxima (respectively, left-to-right minima), and a statistic we call steps. Additionally, Θ (respectively, Ω) fixes all permutations avoiding both 1423 and 2413 (respectively, 3214 and 2413).

Extending the notion of pattern avoidance in permutations, we study matchings and set partitions whose arc diagram representation avoids a given configuration of three arcs. These configurations, which generalize 3-crossings and 3-nestings, have an interpretation, in the case of matchings, in terms of patterns in full rook placements on Ferrers boards. We enumerate 312-avoiding matchings and partitions, obtaining algebraic generating functions, unlike in the 321-avoiding (i.e., 3-noncrossing) case. Our approach also provides a more direct proof of a formula of Bóna for the number of 1342-avoiding permutations. Additionally, we give a bijection proving the shape-Wilf-equivalence of the patterns 321 and 213 which simplifies existing proofs by Backelin–West–Xin and Jelínek.

Extending the notion of pattern avoidance in permutations, we study matchings and set partitions whose arc diagram representation avoids a given configuration of three arcs. These configurations, which generalize 3-crossings and 3-nestings, have an interpretation, in the case of matchings, in terms of patterns in full rook placements on Ferrers boards. We enumerate 312-avoiding matchings and partitions, obtaining algebraic generating functions, in contrast with the known D-finite generating functions for the 321-avoiding (i.e., 3-noncrossing) case. Our approach also provides a more direct proof of a formula of B\'ona for the number of 1342-avoiding permutations. Additionally, we give a bijection proving the shape-Wilf-equivalence of the patterns 321 and 213 which greatly simplifies existing proofs by Backelin--West--Xin and Jel\'{\i}nek, and provides an extension of work of Gouyou-Beauchamps for matchings with fixed points. Finally, we classify pairs of patterns of length 3 according to shape-Wilf-equivalence, and enumerate matchings and partitions avoiding a pair in most of the resulting equivalence classes.

In their paper on Wilf-equivalence for singleton classes, Backelin, West, and Xin introduced a transformation ϕ⁎, defined by an iterative process and operating on (all) full rook placements on Ferrers boards. Bousquet-Mélou and Steingrímsson proved the analogue of the main result of Backelin, West, and Xin in the context of involutions, and in so doing they needed to prove that ϕ⁎ commutes with the operation of taking inverses. The proof of this commutation result was long and difficult, and Bousquet-Mélou and Steingrímsson asked if ϕ⁎ might be reformulated in such a way as to make this result obvious. In the present paper we provide such a reformulation of ϕ⁎, by modifying the growth diagram algorithm of Fomin. This also answers a question of Krattenthaler, who noted that a bijection defined by the unmodified Fomin algorithm obviously commutes with inverses, and asked what the connection is between this bijection and ϕ⁎.

Stankova and West proved in 2002 that the patterns 231 and 312 are shape-Wilf-equivalent. Their proof was nonbijective and fairly complicated. We give a new characterization of 231 and 312 avoiding full rook placements and use this to give a simple bijective proof of the shape-Wilf- equivalence.

In Bloom and Saracino (2009) [2] we proved that a natural bijection Γ:Sn(321)→Sn(132) that Robertson defined by an iterative process in Robertson (2004) [8] preserves the numbers of fixed points and excedances in each σ∈Sn(321). The proof depended on first showing that Γ(σ−1)=(Γ(σ))−1 for all σ∈Sn(321). Here we give a noniterative definition of Γ that frees the result about fixed points and excedances from its dependence on the result about inverses, while also greatly simplifying and elucidating the result about inverses. We also establish a simple connection between Γ and an analogous bijection ϕ∗:Sn(213)→Sn(321) introduced in Backelin et al. (2007) [1] and studied in Bousquet-Melou and Steingrimsson (2005) [3].