2,709 research outputs found
Labeled Partitions with Colored Permutations
In this paper, we extend the notion of labeled partitions with ordinary
permutations to colored permutations in the sense that the colors are endowed
with a cyclic structure. We use labeled partitions with colored permutations to
derive the generating function of the indices of colored
permutations. The second result is a combinatorial treatment of a relation on
the q-derangement numbers with respect to colored permutations which leads to
the formula of Chow for signed permutations and the formula of Faliharimalala
and Zeng [10] on colored permutations. The third result is an involution on
permutations that implies the generating function formula for the signed
q-counting of the major indices due to Gessel and Simon. This involution can be
extended to signed permutations. In this way, we obtain a combinatorial
interpretation of a formula of Adin, Gessel and Roichman.Comment: 14 page
Bijective enumeration of some colored permutations given by the product of two long cycles
Let be the permutation on symbols defined by $\gamma_n = (1\
2\...\ n)\betannp\gamma_n \beta^{-1}\frac{1}{n- p+1}\alpha\gamma_n\alphamn+1$, an
unexpected connection previously found by several authors by means of algebraic
methods. Moreover, our bijection allows us to refine the latter result with the
cycle type of the permutations.Comment: 22 pages. Version 1 is a short version of 12 pages, entitled "Linear
coefficients of Kerov's polynomials: bijective proof and refinement of
Zagier's result", published in DMTCS proceedings of FPSAC 2010, AN, 713-72
Crossings and nestings in colored set partitions
Chen, Deng, Du, Stanley, and Yan introduced the notion of -crossings and
-nestings for set partitions, and proved that the sizes of the largest
-crossings and -nestings in the partitions of an -set possess a
symmetric joint distribution. This work considers a generalization of these
results to set partitions whose arcs are labeled by an -element set (which
we call \emph{-colored set partitions}). In this context, a -crossing or
-nesting is a sequence of arcs, all with the same color, which form a
-crossing or -nesting in the usual sense. After showing that the sizes of
the largest crossings and nestings in colored set partitions likewise have a
symmetric joint distribution, we consider several related enumeration problems.
We prove that -colored set partitions with no crossing arcs of the same
color are in bijection with certain paths in \NN^r, generalizing the
correspondence between noncrossing (uncolored) set partitions and 2-Motzkin
paths. Combining this with recent work of Bousquet-M\'elou and Mishna affords a
proof that the sequence counting noncrossing 2-colored set partitions is
P-recursive. We also discuss how our methods extend to several variations of
colored set partitions with analogous notions of crossings and nestings.Comment: 25 pages; v2: material revised and condensed; v3 material further
revised, additional section adde
Triangle-Free Triangulations, Hyperplane Arrangements and Shifted Tableaux
Flips of diagonals in colored triangle-free triangulations of a convex
polygon are interpreted as moves between two adjacent chambers in a certain
graphic hyperplane arrangement. Properties of geodesics in the associated flip
graph are deduced. In particular, it is shown that: (1) every diagonal is
flipped exactly once in a geodesic between distinguished pairs of antipodes;
(2) the number of geodesics between these antipodes is equal to twice the
number of Young tableaux of a truncated shifted staircase shape.Comment: figure added, plus several minor change
Bijective Enumeration of 3-Factorizations of an N-Cycle
This paper is dedicated to the factorizations of the symmetric group.
Introducing a new bijection for partitioned 3-cacti, we derive an el- egant
formula for the number of factorizations of a long cycle into a product of
three permutations. As the most salient aspect, our construction provides the
first purely combinatorial computation of this number
Continued fractions for permutation statistics
We explore a bijection between permutations and colored Motzkin paths that
has been used in different forms by Foata and Zeilberger, Biane, and Corteel.
By giving a visual representation of this bijection in terms of so-called cycle
diagrams, we find simple translations of some statistics on permutations (and
subsets of permutations) into statistics on colored Motzkin paths, which are
amenable to the use of continued fractions. We obtain new enumeration formulas
for subsets of permutations with respect to fixed points, excedances, double
excedances, cycles, and inversions. In particular, we prove that cyclic
permutations whose excedances are increasing are counted by the Bell numbers.Comment: final version formatted for DMTC
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