87 research outputs found
Trees, Parking Functions and Factorizations of Full Cycles
Parking functions of length are well known to be in correspondence with
both labelled trees on vertices and factorizations of the full cycle
into transpositions. In fact, these
correspondences can be refined: Kreweras equated the area enumerator of parking
functions with the inversion enumerator of labelled trees, while an elegant
bijection of Stanley maps the area of parking functions to a natural statistic
on factorizations of . We extend these relationships in two principal
ways. First, we introduce a bivariate refinement of the inversion enumerator of
trees and show that it matches a similarly refined enumerator for
factorizations. Secondly, we characterize all full cycles such that
Stanley's function remains a bijection when the canonical cycle is
replaced by . We also exhibit a connection between our refined
inversion enumerator and Haglund's bounce statistic on parking functions.Comment: 23 pages, 8 figure
Minimal factorizations of permutations into star transpositions
We give a compact expression for the number of factorizations of any permutation into a minimal number of transpositions of the form (excluded due to format error) source. This generalizes earlier work of Pak in which substantial restrictions were placed on the permutation being factored. Our result exhibits an unexpected and simple symmetry of star factorizations that has yet to be explained in a satisfactory manner
Tree-like properties of cycle factorizations
We provide a bijection between the set of factorizations, that is, ordered
(n-1)-tuples of transpositions in whose product is (12...n),
and labelled trees on vertices. We prove a refinement of a theorem of
D\'{e}nes that establishes new tree-like properties of factorizations. In
particular, we show that a certain class of transpositions of a factorization
correspond naturally under our bijection to leaf edges of a tree. Moreover, we
give a generalization of this fact.Comment: 10 pages, 3 figure
Parking functions, tree depth and factorizations of the full cycle into transpositions
International audienceConsider the set Fn of factorizations of the full cycle (0 1 2 · · · n) ∈ S{0,1,...,n} into n transpositions. Write any such factorization (a1 b1) · · · (an bn) with all ai < bi to define its lower and upper sequences (a1, . . . , an) and (b1,...,bn), respectively. Remarkably, any factorization can be uniquely recovered from its lower (or upper) sequence. In fact, Biane (2002) showed that the simple map sending a factorization to its lower sequence is a bijection from Fn to the set Pn of parking functions of length n. Reversing this map to recover the factorization (and, hence, upper sequence) corresponding to a given lower sequence is nontrivial
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