14 research outputs found
A Schur function identity related to the (-1)-enumeration of self-complementary plane partitions
We give another proof for the (-1)-enumeration of self-complementary plane
partitions with at least one odd side-length by specializing a certain Schur
function identity. The proof is analogous to Stanley's proof for the ordinary
enumeration. In addition, we obtain enumerations of 180-degree symmetric
rhombus tilings of hexagons with a barrier of arbitrary length along the
central line.Comment: AMSLatex, 14 pages, Parity conditions in Theorem 3 corrected and an
additional case adde
Domino tilings with barriers
In this paper, we continue the study of domino-tilings of Aztec diamonds. In
particular, we look at certain ways of placing ``barriers'' in the Aztec
diamond, with the constraint that no domino may cross a barrier. Remarkably,
the number of constrained tilings is independent of the placement of the
barriers. We do not know of a combinatorial explanation of this fact; our proof
uses the Jacobi-Trudi identity.Comment: 5 pages (two-column format), 1 figur
Bijective proofs for Schur function identities which imply Dodgson's condensation formula and Pl\"ucker relations
We present a ``method'' for bijective proofs for determinant identities,
which is based on translating determinants to Schur functions by the
Jacobi--Trudi identity. We illustrate this ``method'' by generalizing a
bijective construction (which was first used by Goulden) to a class of Schur
function identities, from which we shall obtain bijective proofs for Dodgson's
condensation formula, Pl\"ucker relations and a recent identity of the second
author.Comment: Co-author Michael Kleber added a new proof of his theorem by
inclusion-exclusio
Spin invariant theory for the symmetric group
We formulate a theory of invariants for the spin symmetric group in some
suitable modules which involve the polynomial and exterior algebras. We solve
the corresponding graded multiplicity problem in terms of specializations of
the Schur Q-functions and a shifted q-hook formula. In addition, we provide a
bijective proof for a formula of the principal specialization of the Schur
Q-functions.Comment: v2, 17 pages, modified Introduction and updated references. v3,
correction of typos, final versio