130 research outputs found
Symmetry classes of alternating-sign matrices under one roof
In a previous article [math.CO/9712207], we derived the alternating-sign
matrix (ASM) theorem from the Izergin-Korepin determinant for a partition
function for square ice with domain wall boundary. Here we show that the same
argument enumerates three other symmetry classes of alternating-sign matrices:
VSASMs (vertically symmetric ASMs), even HTSASMs (half-turn-symmetric ASMs),
and even QTSASMs (quarter-turn-symmetric ASMs). The VSASM enumeration was
conjectured by Mills; the others by Robbins [math.CO/0008045]. We introduce
several new types of ASMs: UASMs (ASMs with a U-turn side), UUASMs (two U-turn
sides), OSASMs (off-diagonally symmetric ASMs), OOSASMs (off-diagonally,
off-antidiagonally symmetric), and UOSASMs (off-diagonally symmetric with
U-turn sides). UASMs generalize VSASMs, while UUASMs generalize VHSASMs
(vertically and horizontally symmetric ASMs) and another new class, VHPASMs
(vertically and horizontally perverse). OSASMs, OOSASMs, and UOSASMs are
related to the remaining symmetry classes of ASMs, namely DSASMs (diagonally
symmetric), DASASMs (diagonally, anti-diagonally symmetric), and TSASMs
(totally symmetric ASMs). We enumerate several of these new classes, and we
provide several 2-enumerations and 3-enumerations.
Our main technical tool is a set of multi-parameter determinant and Pfaffian
formulas generalizing the Izergin-Korepin determinant for ASMs and the Tsuchiya
determinant for UASMs [solv-int/9804010]. We evaluate specializations of the
determinants and Pfaffians using the factor exhaustion method.Comment: 16 pages, 16 inline figures. Introduction rewritten with more
motivation and context. To appear in the Annals of Mathematic
Signed Lozenge Tilings
It is well-known that plane partitions, lozenge tilings of a hexagon, perfect matchings on a honeycomb graph, and families of non-intersecting lattice paths in a hexagon are all in bijection. In this work we consider regions that are more general than hexagons. They are obtained by further removing upward-pointing triangles. We call the resulting shapes triangular regions. We establish signed versions of the latter three bijections for triangular regions. We first investigate the tileability of triangular regions by lozenges. Then we use perfect matchings and families of non-intersecting lattice paths to define two signs of a lozenge tiling. Using a new method that we call resolution of a puncture, we show that the two signs are in fact equivalent. As a consequence, we obtain the equality of determinants, up to sign, that enumerate signed perfect matchings and signed families of lattice paths of a triangular region, respectively. We also describe triangular regions, for which the signed enumerations agree with the unsigned enumerations
The permutation classes Av(1234,2341) and Av(1243,2314)
We investigate the structure of the two permutation classes defined by the sets of forbidden patterns \{{1234, 2341}\} and \{{1243, 2314}\}. By considering how the Hasse graphs of permutations in these classes can be built from a sequence of rooted source graphs, we determine their algebraic generating functions. Our approach is similar to that of “adding a slice”, used previously to enumerate various classes of polyominoes and other combinatorial structures. To solve the relevant functional equations, we make extensive use of the kernel method
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