29 research outputs found
Pseudo-centrosymmetric matrices, with applications to counting perfect matchings
We consider square matrices A that commute with a fixed square matrix K, both
with entries in a field F not of characteristic 2. When K^2=I, Tao and Yasuda
defined A to be generalized centrosymmetric with respect to K. When K^2=-I, we
define A to be pseudo-centrosymmetric with respect to K; we show that the
determinant of every even-order pseudo-centrosymmetric matrix is the sum of two
squares over F, as long as -1 is not a square in F. When a
pseudo-centrosymmetric matrix A contains only integral entries and is
pseudo-centrosymmetric with respect to a matrix with rational entries, the
determinant of A is the sum of two integral squares. This result, when
specialized to when K is the even-order alternating exchange matrix, applies to
enumerative combinatorics. Using solely matrix-based methods, we reprove a weak
form of Jockusch's theorem for enumerating perfect matchings of 2-even
symmetric graphs. As a corollary, we reprove that the number of domino tilings
of regions known as Aztec diamonds and Aztec pillows is a sum of two integral
squares.Comment: v1: Preprint; 11 pages, 7 figures. v2: Preprint; 15 pages, 7 figures.
Reworked so that linear algebraic results are over a field not of
characteristic 2, not over the real numbers. Accepted, Linear Algebra and its
Application
Abacus models for parabolic quotients of affine Weyl groups
We introduce abacus diagrams that describe minimal length coset
representatives in affine Weyl groups of types B, C, and D. These abacus
diagrams use a realization of the affine Weyl group of type C due to Eriksson
to generalize a construction of James for the symmetric group. We also describe
several combinatorial models for these parabolic quotients that generalize
classical results in affine type A related to core partitions.Comment: 28 pages, To appear, Journal of Algebra. Version 2: Updated with
referee's comment
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
A -Queens Problem. II. The Square Board
We apply to the chessboard the counting theory from Part I for
nonattacking placements of chess pieces with unbounded straight-line moves,
such as the queen. Part I showed that the number of ways to place identical
nonattacking pieces is given by a quasipolynomial function of of degree
, whose coefficients are (essentially) polynomials in that depend
cyclically on .
Here we study the periods of the quasipolynomial and its coefficients, which
are bounded by functions, not well understood, of the piece's move directions,
and we develop exact formulas for the very highest coefficients. The
coefficients of the three highest powers of do not vary with . On the
other hand, we present simple pieces for which the fourth coefficient varies
periodically. We develop detailed properties of counting quasipolynomials that
will be applied in sequels to partial queens, whose moves are subsets of those
of the queen, and the nightrider, whose moves are extended knight's moves.
We conclude with the first, though strange, formula for the classical
-Queens Problem and with several conjectures and open problems.Comment: 23 pp., 1 figure, submitted. This = second half of 1303.1879v1 with
great improvements. V2 has a new proposition, better definitions, and
corrected conjectures. V3 has results et al. renumbered to correspond with
published version, and expands dictionary's cryptic abbreviation