Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userBlock decomposition and statistics arising from permutation tableaux
95
0
0.0
(
0
)
Authors
Joanna N. Chen
Ask ChatGPT about the research
No Arabic abstract
Permutation statistics $wnm$ and $rlm$ are both arising from permutation tableaux. $wnm$ was introduced by Chen and Zhou, which was proved equally distributed with the number of unrestricted rows of a permutation tableau. While $rlm$ is showed by Nadeau equally distributed with the number of $1$s in the first row of a permutation tableau. In this paper, we investigate the joint distribution of $wnm$ and $rlm$. Statistic $(rlm,wnm,rlmin,des,(underline{321}))$ is shown equally distributed with $(rlm,rlmin,wnm,des,(underline{321}))$ on $S_n$. Then the generating function of $(rlm,wnm)$ follows. An involution is constructed to explain the symmetric property of the generating function. Also, we study the triple statistic $(wnm,rlm,asc)$, which is shown to be equally distributed with $(rlmax-1,rlmin,asc)$ as studied by Josuat-Verg$grave{e}$s. The main method we adopt throughout the paper is constructing bijections based on a block decomposition of permutations.
rate research
Read More
Permutation tableaux were introduced by Steingr{i}msson and Williams. Corteel and Kim defined the sign of a permutation tableau in terms of the number of unrestricted columns. The sign-imbalance of permutation tableaux of length $n$ is the sum of signs over permutation tableaux of length $n$. They have btained a formula for the sign-imbalance of permutation tableaux of length $n$ by using generating functions and asked for a combinatorial proof. Moreover, they raised the question of finding a sign-imbalance formula for type $B$ permutation tableaux introduced by Lam and Williams. We define a statistic $ wnm$ over permutations and show that the number of unrestricted columns over permutation tableaux of length $n$ is equally distributed with $ wnm$ over permutations of length $n$. This leads to a combinatorial interpretation of the formula of Corteel and Kim. For type $B$ permutation tableaux, we define the sign of a type $B$ permutation tableau in term of the number of certain rows and columns. On the other hand, we construct a bijection between the type $B$ permutation tableaux of length $n$ and symmetric permutations of length $2n$ and we show that the statistic $ wnm$ over symmetric permutations of length $2n$ is equally distributed with the number of certain rows and columns over type $B$ permutation tableaux of length $n$. Based on this correspondence and an involution on symmetric permutation of length $2n$, we obtain a sign-imbalance formula for type $B$ permutation tableaux.
We define an excedance number for the multi-colored permutation group, i.e. the wreath product of Z_{r_1} x ... x Z_{r_k} with S_n, and calculate its multi-distribution with some natural parameters. We also compute the multi-distribution of the parameters exc(pi) and fix(pi) over the sets of involutions in the multi-colored permutation group. Using this, we count the number of involutions in this group having a fixed number of excedances and absolute fixed points.
In 1976, King defined certain tableaux model, called King tableaux in this paper, counting weight multiplicities of irreducible representation of the symplectic group $Sp(2m)$ for a given dominant weight. Since Kashiwara defined crystals, it is an open problem to provide a crystal structure on King tableaux. In this paper, we present crystal structures on King tableaux and semistandard oscillating tableaux. The semistandard oscillating tableaux naturally appear as $Q$-tableaux in the symplectic version of RSK algorithms. As an application, we discuss Littlewood-Richardson coefficients for $Sp(2m)$ in terms of semistandard oscillating tableaux.
We give a counting formula for the set of rectangular increasing tableaux in terms of generalized Narayana numbers. We define small $m$-Schroder paths and give a bijection between the set of increasing rectangular tableaux and small $m$-Schroder paths, generalizing a result of Pechenik [3]. Using $K$-jeu de taquin promotion, which was defined by Thomas and Yong [10], we give a cyclic sieving phenomenon for the set of increasing hook tableaux.
As shown by Bousquet-Melou--Claesson--Dukes--Kitaev (2010), ascent sequences can be used to encode $({bf2+2})$-free posets. It is known that ascent sequences are enumerated by the Fishburn numbers, which appear as the coefficients of the formal power series $$sum_{m=1}^{infty}prod_{i=1}^m (1-(1-t)^i).$$ In this paper, we present a novel way to recursively decompose ascent sequences, which leads to: (i) a calculation of the Euler--Stirling distribution on ascent sequences, including the numbers of ascents ($asc$), repeated entries $(rep)$, zeros ($zero$) and maximal entries ($max$). In particular, this confirms and extends Dukes and Parviainens conjecture on the equidistribution of $zero$ and $max$. (ii) a far-reaching generalization of the generating function formula for $(asc,zero)$ due to Jelinek. This is accomplished via a bijective proof of the quadruple equidistribution of $(asc,rep,zero,max)$ and $(rep,asc,rmin,zero)$, where $rmin$ denotes the right-to-left minima statistic of ascent sequences. (iii) an extension of a conjecture posed by Levande, which asserts that the pair $(asc,zero)$ on ascent sequences has the same distribution as the pair $(rep,max)$ on $({bf2-1})$-avoiding inversion sequences. This is achieved via a decomposition of $({bf2-1})$-avoiding inversion sequences parallel to that of ascent sequences. This work is motivated by a double Eulerian equidistribution of Foata (1977) and a tempting bi-symmetry conjecture, which asserts that the quadruples $(asc,rep,zero,max)$ and $(rep,asc,max,zero)$ are equidistributed on ascent sequences.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
