相关论文: Involutions under Bruhat order and labeled Motzkin…

We show how a bijection due to Biane between involutions and labelled Motzkin paths yields bijections between Motzkin paths and two families of restricted involutions that are counted by Motzkin numbers, namely, involutions avoiding 4321…

Laguerre histories (restricted or not) are certain weighted Motzkin paths with two types of level steps. They are, on one hand, in natural bijection with the set of permutations, and on the other hand, yield combinatorial interpretations…

In their paper [1] on Wilf-equivalence for singleton classes, Backelin, Xin, and West introduce a transformation $\phi^*$, defined by an iterative process and operating on (all) full rook placements on Ferrers boards. In [3],…

The class of ranked tree-child networks, tree-child networks arising from an evolution process with a fixed embedding into the plane, has recently been introduced by Bienvenu, Lambert, and Steel. These authors derived counting results for…

The problem of counting plane trees with $n$ edges and an even or an odd number of leaves was studied by Eu, Liu and Yeh, in connection with an identity on coloring nets due to Stanley. This identity was also obtained by Bonin, Shapiro and…

We construct a bijection from permutations to some weighted Motzkin paths known as Laguerre histories. As one application of our bijection, a neat $q$-$\gamma$-positivity expansion of the $(\inv,\exc)$-$q$-Eulerian polynomials is obtained.

Applying a classical theorem of Smith, we show that the poset property of being Gorenstein$^*$ over $\mathbb{Z}_2$ is inherited by the subposet of fixed points under an involutive poset automorphism. As an application, we prove that every…

We present a simplified variant of Biane's bijection between permutations and 3-colored Motzkin paths with weight that keeps track of the inversion number, excedance number and a statistic so-called depth of a permutation. This generalizes…

We give several bijections among restricted Motzkin paths, explaining why various parameters on these paths are equidistributed. For example, the number of doublerise-free Motzkin paths of length n is the same as the number of peak-free…

The orbits of the orthogonal and symplectic groups on the flag variety are in bijection, respectively, with the involutions and fixed-point-free involutions in the symmetric group $S_n$. Wyser and Yong have described polynomial…

Let $I_n$ be the set of involutions in the symmetric group $S_n$, and for $A \subseteq \{0,1,\ldots,n\}$, let \[ F_n^A=\{\sigma \in I_n \mid \text{$\sigma$ has $a$ fixed points for some $a \in A$}\}. \] We give a complete characterisation…

We derive formulae for the number of set-valued standard tableaux of two-rowed shapes, keeping track of the total number of entries, the number of entries in the first row, and the number of entries in the second row. Key in the proofs is a…

When $Sp(2n,\mathbb{C})$ acts on the flag variety of $SL(2n,\mathbb{C})$, the orbits are in bijection with fixed point free involutions in the symmetric group $S_{2n}$. In this case, the associated Kazhdan-Lusztig-Vogan polynomials…

We consider the framework of an operator-valued noncommutative probability space over a unital C*-algebra B. We show how for a B-valued distribution \mu one can define convolution powers with respect to free additive convolution and with…

We consider the involutions known as "toggles," which have been used to give simplified proofs of the fundamental properties of the promotion and evacuation maps. We transfer these involutions so that they generate a group $\mathscr P_n$…

We explore a bijection between permutations and colored Motzkin paths that has been used in different forms by Foata and Zeilberger, Biane, and Corteel. By giving a visual representation of this bijection in terms of so-called cycle…

We show that the principal order ideal below an element w in the Bruhat order on involutions in a symmetric group is a Boolean lattice if and only if w avoids the patterns 4321, 45312 and 456123. Similar criteria for signed permutations are…

We exhibit a bijection between 132-avoiding permutations and Dyck paths. Using this bijection, it is shown that all the recently discovered results on generating functions for 132-avoiding permutations with a given number of occurrences of…

We introduce bijections between families of rooted maps with unfixed genus and families of so-called blossoming trees endowed with an arbitrary forward matching of their leaves. We first focus on Eulerian maps with controlled vertex…

Suppose the rows of a board are partitioned into sets of m rows called levels. An m-level rook placement is a subset of the board where no two squares are in the same column or the same level. We construct explicit bijections to prove three…