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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
Egge triples and unbalanced Wilf-equivalence
Egge conjectured that permutations avoiding the set of patterns
, where ,
are enumerated by the large Schr\"oder numbers. Consequently,
with as above is Wilf-equivalent to the set of
patterns . Burstein and Pantone proved the case of
. We prove the remaining four cases. As a byproduct of our proof,
we also enumerate the case .Comment: 20 pages, 6 figures (published version
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