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

Proofs of two conjectures of Kenyon and Wilson on Dyck tilings

Recently, Kenyon and Wilson introduced a certain matrix $M$ in order to compute pairing probabilities of what they call the double-dimer model. They showed that the absolute value of each entry of the inverse matrix $M^{-1}$ is equal to the number of certain Dyck tilings of a skew shape. They conjectured two formulas on the sum of the absolute values of the entries in a row or a column of $M^{-1}$. In this paper we prove the two conjectures. As a consequence we obtain that the sum of the absolute values of all entries of $M^{-1}$ is equal to the number of complete matchings. We also find a bijection between Dyck tilings and complete matchings.Comment: 18 pages, 9 figure