Growth diagrams, and increasing and decreasing chains in fillings of Ferrers shapes

We put recent results by Chen, Deng, Du, Stanley and Yan on crossings and nestings of matchings and set partitions in the larger context of the enumeration of fillings of Ferrers shape on which one imposes restrictions on their increasing and decreasing chains. While Chen et al. work with Robinson-Schensted-like insertion/deletion algorithms, we use the growth diagram construction of Fomin to obtain our results. We extend the results by Chen et al., which, in the language of fillings, are results about $0$-$1$-fillings, to arbitrary fillings. Finally, we point out that, very likely, these results are part of a bigger picture which also includes recent results of Jonsson on $0$-$1$-fillings of stack polyominoes, and of results of Backelin, West and Xin and of Bousquet-M\'elou and Steingr\'\i msson on the enumeration of permutations and involutions with restricted patterns. In particular, we show that our growth diagram bijections do in fact provide alternative proofs of the results by Backelin, West and Xin and by Bousquet-M\'elou and Steingr\'\i msson.Comment: AmS-LaTeX; 27 pages; many corrections and improvements of short-comings; thanks to comments by Mireille Bousquet-Melou and Jakob Jonsson, the final section is now much more profound and has additional result

Partition Statistics Equidistributed with the Number of Hook Difference One Cells

Let $\lambda$ be a partition, viewed as a Young diagram. We define the hook difference of a cell of $\lambda$ to be the difference of its leg and arm lengths. Define $h_{1,1}(\lambda)$ to be the number of cells of $\lambda$ with hook difference one. In the paper of Buryak and Feigin (arXiv:1206.5640), algebraic geometry is used to prove a generating function identity which implies that $h_{1,1}$ is equidistributed with $a_2$, the largest part of a partition that appears at least twice, over the partitions of a given size. In this paper, we propose a refinement of the theorem of Buryak and Feigin and prove some partial results using combinatorial methods. We also obtain a new formula for the q-Catalan numbers which naturally leads us to define a new q,t-Catalan number with a simple combinatorial interpretation

State complexity of catenation combined with a boolean operation: a unified approach

In this paper we study the state complexity of catenation combined with symmetric difference. First, an upper bound is computed using some combinatoric tools. Then, this bound is shown to be tight by giving a witness for it. Moreover, we relate this work with the study of state complexity for two other combinations: catenation with union and catenation with intersection. And we extract a unified approach which allows to obtain the state complexity of any combination involving catenation and a binary boolean operation

Enumerations of Permutations Simultaneously Avoiding a Vincular and a Covincular Pattern of Length 3

Vincular and covincular patterns are generalizations of classical patterns allowing restrictions on the indices and values of the occurrences in a permutation. In this paper we study the integer sequences arising as the enumerations of permutations simultaneously avoiding a vincular and a covincular pattern, both of length 3, with at most one restriction. We see familiar sequences, such as the Catalan and Motzkin numbers, but also some previously unknown sequences which have close links to other combinatorial objects such as lattice paths and integer partitions. Where possible we include a generating function for the enumeration. One of the cases considered settles a conjecture by Pudwell (2010) on the Wilf-equivalence of barred patterns. We also give an alternative proof of the classic result that permutations avoiding 123 are counted by the Catalan numbers.Comment: 24 pages, 11 figures, 2 table

Frame patterns in n-cycles

In this paper, we study the distribution of the number of occurrences of the simplest frame pattern, called the $\mu$ pattern, in $n$-cycles. Given an $n$-cycle $C$, we say that a pair $\langle i,j \rangle$ matches the $\mu$ pattern if $i < j$ and as we traverse around $C$ in a clockwise direction starting at $i$ and ending at $j$, we never encounter a $k$ with $i < k < j$. We say that $\langle i,j \rangle$ is a nontrivial $\mu$-match if $i+1 < j$. Also, an $n$-cycle $C$ is incontractible if there is no $i$ such that $i+1$ immediately follows $i$ in $C$. We show that the number of incontractible $n$-cycles in the symmetric group $S_n$ is $D_{n-1}$, where $D_n$ is the number of derangements in $S_n$. Further, we prove that the number of $n$-cycles in $S_n$ with exactly $k$ $\mu$-matches can be expressed as a linear combination of binomial coefficients of the form $\binom{n-1}{i}$ where $i \leq 2k+1$. We also show that the generating function $NTI_{n,\mu}(q)$ of $q$ raised to the number of nontrivial $\mu$-matches in $C$ over all incontractible $n$-cycles in $S_n$ is a new $q$-analogue of $D_{n-1}$, which is different from the $q$-analogues of the derangement numbers that have been studied by Garsia and Remmel and by Wachs. We show that there is a rather surprising connection between the charge statistic on permutations due to Lascoux and Sch\"uzenberger and our polynomials in that the coefficient of the smallest power of $q$ in $NTI_{2k+1,\mu}(q)$ is the number of permutations in $S_{2k+1}$ whose charge path is a Dyck path. Finally, we show that $NTI_{n,\mu}(q)|_{q^{\binom{n-1}{2} -k}}$ and $NT_{n,\mu}(q)|_{q^{\binom{n-1}{2} -k}}$ are the number of partitions of $k$ for sufficiently large $n$