61 research outputs found
Hook formulas for skew shapes III. Multivariate and product formulas
We give new product formulas for the number of standard Young tableaux of
certain skew shapes and for the principal evaluation of the certain Schubert
polynomials. These are proved by utilizing symmetries for evaluations of
factorial Schur functions, extensively studied in the first two papers in the
series "Hook formulas for skew shapes" [arxiv:1512.08348, arxiv:1610.04744]. We
also apply our technology to obtain determinantal and product formulas for the
partition function of certain weighted lozenge tilings, and give various
probabilistic and asymptotic applications.Comment: 40 pages, 17 figures. This is the third paper in the series "Hook
formulas for skew shapes"; v2 added reference to [KO1] (arxiv:1409.1317)
where the formula in Corollary 1.1 had previously appeared; v3 Corollary 5.10
added, resembles published versio
The enumeration of permutations avoiding 2143 and 4231
We enumerate the pattern class Av(2143, 4231) and completely describe its permutations. The main tools are simple permutations and monotone grid classes
Finitely labeled generating trees and restricted permutations
Generating trees are a useful technique in the enumeration of various
combinatorial objects, particularly restricted permutations. Quite often the
generating tree for the set of permutations avoiding a set of patterns requires
infinitely many labels. Sometimes, however, this generating tree needs only
finitely many labels. We characterize the finite sets of patterns for which
this phenomenon occurs. We also present an algorithm - in fact, a special case
of an algorithm of Zeilberger - that is guaranteed to find such a generating
tree if it exists.Comment: Accepted by J. Symb. Comp.; 12 page
- …