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Partition Statistics Equidistributed with the Number of Hook Difference One Cells
Let be a partition, viewed as a Young diagram. We define the hook
difference of a cell of to be the difference of its leg and arm
lengths. Define to be the number of cells of with
hook difference one. In the paper of Buryak and Feigin (arXiv:1206.5640),
algebraic geometry is used to prove a generating function identity which
implies that is equidistributed with , the largest part of a
partition that appears at least twice, over the partitions of a given size. In
this paper, we propose a refinement of the theorem of Buryak and Feigin and
prove some partial results using combinatorial methods. We also obtain a new
formula for the q-Catalan numbers which naturally leads us to define a new
q,t-Catalan number with a simple combinatorial interpretation
Identities from representation theory
We give a new Jacobi--Trudi-type formula for characters of finite-dimensional
irreducible representations in type using characters of the fundamental
representations and non-intersecting lattice paths. We give equivalent
determinant formulas for the decomposition multiplicities for tensor powers of
the spin representation in type and the exterior representation in type
. This gives a combinatorial proof of an identity of Katz and equates such
a multiplicity with the dimension of an irreducible representation in type
. By taking certain specializations, we obtain identities for -Catalan
triangle numbers, the -Catalan number of Stump, -triangle versions of
Motzkin and Riordan numbers, and generalizations of Touchard's identity. We use
(spin) rigid tableaux and crystal base theory to show some formulas relating
Catalan, Motzkin, and Riordan triangle numbers.Comment: 68 pages, 8 figure
A combinatorial identity with application to Catalan numbers
By a very simple argument, we prove that if are nonnegative integers
then \sum_{k=0}^l(-1)^{m-k}\binom{l}{k}\binom{m-k}{n}\binom{2k}{k-2l+m}
=\sum_{k=0}^l\binom{l}{k}\binom{2k}{n}\binom{n-l}{m+n-3k-l}.
On the basis of this identity, for we construct explicit
and such that for any prime we have
\sum_{k=1}^{p-1}k^r C_{k+d}\equiv \cases F(d,r)(mod p)& if 3|p-1, \\G(d,r)\
(mod p)& if 3|p-2,
where denotes the Catalan number . For
example, when is a prime, we have
\sum_{k=1}^{p-1}k^2C_k\equiv\cases-2/3 (mod p)& if 3|p-1, \1/3 (mod p)& if
3|p-2;
and
\sum_{0<k<p-4}\frac{C_{k+4}}k \equiv\cases 503/30 (mod p)& if 3|p-1, -100/3
(mod p)& if 3|p-2.
This paper also contains some new recurrence relations for Catalan numbers.Comment: 22 page
A note on super Catalan numbers
We show that the super Catalan numbers are special values of the Krawtchouk
polynomials by deriving an expression for the super Catalan numbers in terms of
a signed set.Comment: 4 pages. Revised and Accepted. To appear in Interdisciplinary
Information Sciences (IIS
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