1,402 research outputs found
A Graph Theory of Rook Placements
Two boards are rook equivalent if they have the same number of non-attacking
rook placements for any number of rooks. Define a rook equivalence graph of an
equivalence set of Ferrers boards by specifying that two boards are connected
by an edge if you can obtain one of the boards by moving squares in the other
board out of one column and into a singe other column. Given such a graph, we
categorize which boards will yield connected graphs. We also provide some cases
where common graphs will or will not be the graph for some set of rook
equivalent Ferrers boards. Finally, we extend this graph definition to the
-level rook placement generalization developed by Briggs and Remmel. This
yields a graph on the set of rook equivalent singleton boards, and we
characterize which singleton boards give rise to a connected graph.Comment: 15 pages, 9 figure
Two Vignettes On Full Rook Placements
Using bijections between pattern-avoiding permutations and certain full rook
placements on Ferrers boards, we give short proofs of two enumerative results.
The first is a simplified enumeration of the 3124, 1234-avoiding permutations,
obtained recently by Callan via a complicated decomposition. The second is a
streamlined bijection between 1342-avoiding permutations and permutations which
can be sorted by two increasing stacks in series, originally due to Atkinson,
Murphy, and Ru\v{s}kuc.Comment: 9 pages, 4 figure
Stammering tableaux
The PASEP (Partially Asymmetric Simple Exclusion Process) is a probabilistic
model of moving particles, which is of great interest in combinatorics, since
it appeared that its partition function counts some tableaux. These tableaux
have several variants such as permutations tableaux, alternative tableaux,
tree- like tableaux, Dyck tableaux, etc. We introduce in this context certain
excursions in Young's lattice, that we call stammering tableaux (by analogy
with oscillating tableaux, vacillating tableaux, hesitating tableaux). Some
natural bijections make a link with rook placements in a double staircase,
chains of Dyck paths obtained by successive addition of ribbons, Laguerre
histories, Dyck tableaux, etc.Comment: Clarification and better exposition thanks reviewer's report
Elliptic rook and file numbers
Utilizing elliptic weights, we construct an elliptic analogue of rook numbers
for Ferrers boards. Our elliptic rook numbers generalize Garsia and Remmel's
q-rook numbers by two additional independent parameters a and b, and a nome p.
These are shown to satisfy an elliptic extension of a factorization theorem
which in the classical case was established by Goldman, Joichi and White and
later was extended to the q-case by Garsia and Remmel. We obtain similar
results for our elliptic analogues of Garsia and Remmel's q-file numbers for
skyline boards. We also provide an elliptic extension of the j-attacking model
introduced by Remmel and Wachs. Various applications of our results include
elliptic analogues of (generalized) Stirling numbers of the first and second
kind, Lah numbers, Abel numbers, and r-restricted versions thereof.Comment: 45 pages; 3rd version shortened (elliptic rook theory for matchings
has been taken out to keep the length of this paper reasonable
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