44 research outputs found
Toss and Spin Juggling State Graphs
We review the state approach to toss juggling and extend the approach to spin
juggling, a new concept. We give connections to current research on random
juggling and describe a professional-level juggling performance that further
demonstrates the state graphs and their research.Comment: 8 pages, 10 figures, to appear in the Proceedings of Bridges 201
Combinatorial approach to generalized Bell and Stirling numbers and boson normal ordering problem
We consider the numbers arising in the problem of normal ordering of
expressions in canonical boson creation and annihilation operators. We treat a
general form of a boson string which is shown to be associated with
generalizations of Stirling and Bell numbers. The recurrence relations and
closed-form expressions (Dobiski-type formulas) are obtained for these
quantities by both algebraic and combinatorial methods. By extensive use of
methods of combinatorial analysis we prove the equivalence of the
aforementioned problem to the enumeration of special families of graphs. This
link provides a combinatorial interpretation of the numbers arising in this
normal ordering problem.Comment: 10 pages, 5 figure
Statistics on ordered partitions of sets
We introduce several statistics on ordered partitions of sets, that is, set
partitions where the blocks are permuted arbitrarily. The distribution of these
statistics is closely related to the q-Stirling numbers of the second kind.
Some of the statistics are generalizations of known statistics on set
partitions, but others are entirely new. All the new ones are sums of two
statistics, inspired by statistics on permutations, where one of the two
statistics is based on a certain partial ordering of the blocks of a partition.Comment: Added a Prologue, as this paper is soon to be published in a journa
Number of cycles in the graph of 312-avoiding permutations
The graph of overlapping permutations is defined in a way analogous to the De
Bruijn graph on strings of symbols. That is, for every permutation there is a directed edge from the
standardization of to the standardization of
. We give a formula for the number of cycles of
length in the subgraph of overlapping 312-avoiding permutations. Using this
we also give a refinement of the enumeration of 312-avoiding affine
permutations and point out some open problems on this graph, which so far has
been little studied.Comment: To appear in the Journal of Combinatorial Theory - Series
Crossings, Motzkin paths and Moments
Kasraoui, Stanton and Zeng, and Kim, Stanton and Zeng introduced certain
-analogues of Laguerre and Charlier polynomials. The moments of these
orthogonal polynomials have combinatorial models in terms of crossings in
permutations and set partitions. The aim of this article is to prove simple
formulas for the moments of the -Laguerre and the -Charlier polynomials,
in the style of the Touchard-Riordan formula (which gives the moments of some
-Hermite polynomials, and also the distribution of crossings in matchings).
Our method mainly consists in the enumeration of weighted Motzkin paths, which
are naturally associated with the moments. Some steps are bijective, in
particular we describe a decomposition of paths which generalises a previous
construction of Penaud for the case of the Touchard-Riordan formula. There are
also some non-bijective steps using basic hypergeometric series, and continued
fractions or, alternatively, functional equations.Comment: 21 page