49,657 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
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
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 -analog of the following result due to McKay, Morse and Wilf:
the probability that a random standard Young tableau of size contains a
fixed standard Young tableau of shape tends to
in the large limit, where is the number of
standard Young tableaux of shape . We also consider the probability
that a random pair of standard Young tableaux of the same shape
contains a fixed pair 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 and type in terms of
noncrossing partitions of type . As an application, we get type-preserving
bijections between noncrossing and nonnesting partitions of type , type
and type which are different from those in the recent work of Fink and
Giraldo. We also define Catalan tableaux of type and type , and find
bijections between them and noncrossing partitions of type and type
respectively.Comment: 21 pages, 15 figures, final versio
A note on the total number of cycles of even and odd permutations
We prove bijectively that the total number of cycles of all even permutations
of and the total number of cycles of all odd permutations
of differ by , which was stated as an open problem by
Mikl\'{o}s B\'{o}na. We also prove bijectively the following more general
identity:
where denotes the number of permutations of with cycles.Comment: 4 pages, 2 figures, final versio
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