A combinatorial approach to the power of 2 in the number of involutions

We provide a combinatorial approach to the largest power of $p$ in the number of permutations $\pi$ with $\pi^p=1$, for a fixed prime number $p$. With this approach, we find the largest power of $2$ in the number of involutions, in the signed sum of involutions and in the numbers of even or odd involutions.Comment: 13 page

Enumeration formulas for generalized q-Euler numbers

We find an enumeration formula for a $(t,q)$-Euler number which is a generalization of the $q$-Euler number introduced by Han, Randrianarivony, and Zeng. We also give a combinatorial expression for the $(t,q)$-Euler number and find another formula when $t=\pm q^r$ for any integer $r$. Special cases of our latter formula include the formula of the $q$-Euler number recently found by Josuat-Verg\`es and Touchard-Riordan's formula.Comment: 21 pages, 12 figure

Bijections on two variations of noncrossing partitions

We find bijections on 2-distant noncrossing partitions, 12312-avoiding partitions, 3-Motzkin paths, UH-free Schr{\"o}der paths and Schr{\"o}der paths without peaks at even height. We also give a direct bijection between 2-distant noncrossing partitions and 12312-avoiding partitions.Comment: 10 pages, 9 figures, final versio

q-analog of tableau containment

We prove a $q$-analog of the following result due to McKay, Morse and Wilf: the probability that a random standard Young tableau of size $n$ contains a fixed standard Young tableau of shape $\lambda\vdash k$ tends to $f^{\lambda}/k!$ in the large $n$ limit, where $f^{\lambda}$ is the number of standard Young tableaux of shape $\lambda$. We also consider the probability that a random pair $(P,Q)$ of standard Young tableaux of the same shape contains a fixed pair $(A,B)$ of standard Young tableaux.Comment: 20 pages, to appear J. Combin. Theory. Ser.

New interpretations for noncrossing partitions of classical types

We interpret noncrossing partitions of type $B$ and type $D$ in terms of noncrossing partitions of type $A$. As an application, we get type-preserving bijections between noncrossing and nonnesting partitions of type $B$, type $C$ and type $D$ which are different from those in the recent work of Fink and Giraldo. We also define Catalan tableaux of type $B$ and type $D$, and find bijections between them and noncrossing partitions of type $B$ and type $D$ respectively.Comment: 21 pages, 15 figures, final versio