431 research outputs found
A Survey of Alternating Permutations
This survey of alternating permutations and Euler numbers includes
refinements of Euler numbers, other occurrences of Euler numbers, longest
alternating subsequences, umbral enumeration of classes of alternating
permutations, and the cd-index of the symmetric group.Comment: 32 pages, 7 figure
Why Delannoy numbers?
This article is not a research paper, but a little note on the history of
combinatorics: We present here a tentative short biography of Henri Delannoy,
and a survey of his most notable works. This answers to the question raised in
the title, as these works are related to lattice paths enumeration, to the
so-called Delannoy numbers, and were the first general way to solve Ballot-like
problems. These numbers appear in probabilistic game theory, alignments of DNA
sequences, tiling problems, temporal representation models, analysis of
algorithms and combinatorial structures.Comment: Presented to the conference "Lattice Paths Combinatorics and Discrete
Distributions" (Athens, June 5-7, 2002) and to appear in the Journal of
Statistical Planning and Inference
Linear versus spin: representation theory of the symmetric groups
We relate the linear asymptotic representation theory of the symmetric groups
to its spin counterpart. In particular, we give explicit formulas which express
the normalized irreducible spin characters evaluated on a strict partition
with analogous normalized linear characters evaluated on the double
partition . We also relate some natural filtration on the usual
(linear) Kerov-Olshanski algebra of polynomial functions on the set of Young
diagrams with its spin counterpart. Finally, we give a spin counterpart to
Stanley formula for the characters of the symmetric groups.Comment: 41 pages. Version 2: new text about non-oriented (but orientable)
map
Combinatorial reciprocity for non-intersecting paths
We prove a combinatorial reciprocity theorem for the enumeration of
non-intersecting paths in a linearly growing sequence of acyclic planar
networks. We explain two applications of this theorem: reciprocity for fans of
bounded Dyck paths, and reciprocity for Schur function evaluations with
repeated values.Comment: 18 pages, 8 figures; v2: final version to appear in "Enumerative
Combinatorics and Applications
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