41,320 research outputs found
A combinatorial approach to the power of 2 in the number of involutions
We provide a combinatorial approach to the largest power of in the number
of permutations with , for a fixed prime number . With this
approach, we find the largest power of 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 -Euler number which is a
generalization of the -Euler number introduced by Han, Randrianarivony, and
Zeng. We also give a combinatorial expression for the -Euler number and
find another formula when for any integer . Special cases of our
latter formula include the formula of the -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 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 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 . In this paper we prove the two conjectures. As a
consequence we obtain that the sum of the absolute values of all entries of
is equal to the number of complete matchings. We also find a bijection
between Dyck tilings and complete matchings.Comment: 18 pages, 9 figure
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