9 research outputs found
Avoidance of Partitions of a Three-element Set
Klazar defined and studied a notion of pattern avoidance for set partitions,
which is an analogue of pattern avoidance for permutations. Sagan considered
partitions which avoid a single partition of three elements. We enumerate
partitions which avoid any family of partitions of a 3-element set as was done
by Simion and Schmidt for permutations. We also consider even and odd set
partitions. We provide enumerative results for set partitions restricted by
generalized set partition patterns, which are an analogue of the generalized
permutation patterns of Babson and Steingr{\'{\i}}msson. Finally, in the spirit
of work done by Babson and Steingr{'{\i}}msson, we will show how these
generalized partition patterns can be used to describe set partition
statistics.Comment: 23 pages, 2 tables, 1 figure, to appear in Advances in Applied
Mathematic
The enumeration of fully commutative affine permutations
We give a generating function for the fully commutative affine permutations
enumerated by rank and Coxeter length, extending formulas due to Stembridge and
Barcucci--Del Lungo--Pergola--Pinzani. For fixed rank, the length generating
functions have coefficients that are periodic with period dividing the rank. In
the course of proving these formulas, we obtain results that elucidate the
structure of the fully commutative affine permutations.Comment: 18 pages; final versio
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