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 containing exactly k 123 patterns in terms of ballot numbers for k<=4, and (iii) to express the number of 123-avoiding n-permutations containing exactly k 132 patterns as a linear combination of powers of 2, also for k<=4. The results strengthen the conjecture that the counts are algebraic for all k.
The number of tree-rooted maps, that is, rooted planar maps with a distinguished spanning tree, of size $n$ is C(n)C(n+1) where C(n)=binomial(2n,n)/(n+1) is the nth Catalan number. We present a (long awaited) simple bijection which explains this result. We prove that our bijection is isomorphic to a former recursive construction on shuffles of parenthesis systems due to Cori, Dulucq and Viennot.
We find generating functions for the number of words avoiding certain patterns or sets of patterns on at most 2 distinct letters and determine which of them are equally avoided. We also find the exact number of words avoiding certain patterns and provide bijective proofs for the resulting formulas.
Recently, Naruse presented a beautiful cancellation-free hook-length formula for skew shapes. The formula involves a sum over objects called excited diagrams, and the term corresponding to each excited diagram has hook lengths in the denominator, like the classical hook-length formula due to Frame, Robinson and Thrall. In this paper, we present a simple bijection that proves an equivalent recursive version of Naruse's result, in the same way that the celebrated hook-walk proof due to Green, Nijenhuis and Wilf gives a bijective (or probabilistic) proof of the hook-length formula for ordinary sh…
Bijective proofs of several conjectures on Jacobi permutations
Research Center for Mathematics and Interdisciplinary Sciences, Shandong University & Frontiers Science Center for Nonlinear Expectations, Ministry of Education（山东大学数学与交叉科学研究中心及教育部非线性预期前沿科学中心）；Department of Mathematics, Zhejiang Normal University（浙江师范大学数学系）
We present recent results on the enumeration of $q$-coloured planar maps, where each monochromatic edge carries a weight $ν$. This is equivalent to weighting each map by its Tutte polynomial, or to solving the $q$-state Potts model on random planar maps. The associated generating function, obtained by Olivier Bernardi and the author, is differentially algebraic. That is, it satisfies a (non-linear) differential equation. The starting point of this result is a functional equation written by Tutte in 1971, which translates into enumerative terms a simple recursive description of planar maps. The…
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 are equal if and only if w avoids the ten patterns 4321, 34512, 45123, 35412, 43512, 45132, 45213, 53412, 45312, and 45231.
We consider unicellular maps, or polygon gluings, of fixed genus. A few years ago the first author gave a recursive bijection transforming unicellular maps into trees, explaining the presence of Catalan numbers in counting formulas for these objects. In this paper, we give another bijection that explicitly describes the "recursive part" of the first bijection. As a result we obtain a very simple description of unicellular maps as pairs made by a plane tree and a permutation-like structure. All the previously known formulas follow as an immediate corollary or easy exercise, thus giving a biject…
A permutation $π$ contains a pattern $σ$ if and only if there is a subsequence in $π$ with its letters are in the same relative order as those in $σ$. Partially ordered patterns (POPs) provide a convenient way to denote patterns in which the relative order of some of the letters does not matter. This paper elucidates connections between the avoidance sets of a few POPs with other combinatorial objects, directly answering five open questions posed by Gao and Kitaev \cite{gao-kitaev-2019}. This was done by thoroughly analysing the avoidance sets and developing recursive algorithms to derive thes…
We find the winning strategy for a class of truncation games played on words. As a consequence of the present author's recent results on some of these games we obtain new formulas for Bernoulli numbers and polynomials of the second kind and a new combinatorial model for the number of connected permutations of given rank. For connected permutations, the decomposition used to find the winning strategy is shown to be bijectively equivalent to King's decomposition, used to recursively generate a transposition Gray code of the connected permutations.
Ascent sequences are sequences of nonnegative integers with restrictions on the size of each letter, depending on the number of ascents preceding it in the sequence. Ascent sequences have recently been related to (2+2)-free posets and various other combinatorial structures. We study pattern avoidance in ascent sequences, giving several results for patterns of lengths up to 4, for Wilf equivalence and for growth rates. We establish bijective connections between pattern avoiding ascent sequences and various other combinatorial objects, in particular with set partitions. We also make a number of …
Comtet introduced the notion of indecomposable permutations in 1972. A permutation is indecomposable if and only if it has no proper prefix which is itself a permutation. Indecomposable permutations were studied in the literature in various contexts. In particular, this notion has been proven to be useful in obtaining non-trivial enumeration and equidistribution results on permutations. In this paper, we give a complete classification of indecomposable permutations avoiding a classical pattern of length 3 or 4, and of indecomposable permutations avoiding a non-consecutive vincular pattern of l…
In [BabStein] 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. In [Kit1] Kitaev considered simultaneous avoidance (multi-avoidance) of two or more 3-patterns with no internal dashes, that is, where the patterns correspond to contiguous subwords in a permutation. There either an explicit or a recursive formula was given for all but one case of simultaneous avoidance of more than two patterns. In this paper we find the exponential generating function for the remaining case…
We study pattern avoidance by combinatorial objects other than permutations, namely by ordered partitions of an integer and by permutations of a multiset. In the former case we determine the generating function explicitly, for integer compositions of n that avoid a given pattern of length 3 and we show that the answer is the same for all such patterns. We also show that the number of multiset permutations that avoid a given three-letter pattern is the same for all such patterns, thereby extending and refining earlier results of Albert, Aldred et al., and by Atkinson, Walker and Linton. Further…
It is well-known, and was first established by Knuth in 1969, that the number of 321-avoiding permutations is equal to that of 132-avoiding permutations. In the literature one can find many subsequent bijective proofs of this fact. It turns out that some of the published bijections can easily be obtained from others. In this paper we describe all bijections we were able to find in the literature and show how they are related to each other via ``trivial'' bijections. We classify the bijections according to statistics preserved (from a fixed, but large, set of statistics), obtaining substantial …
We give a recursive formula for the Moebius function of an interval $[σ,π]$ in the poset of permutations ordered by pattern containment in the case where $π$ is a decomposable permutation, that is, consists of two blocks where the first one contains all the letters 1, 2, ..., k for some k. This leads to many special cases of more explicit formulas. It also gives rise to a computationally efficient formula for the Moebius function in the case where $σ$ and $π$ are separable permutations. A permutation is separable if it can be generated from the permutation 1 by successive sums and skew sums or…
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…
