Geometry and complexity of O'Hara's algorithm

In this paper we analyze O'Hara's partition bijection. We present three type of results. First, we show that O'Hara's bijection can be viewed geometrically as a certain scissor congruence type result. Second, we obtain a number of new complexity bounds, proving that O'Hara's bijection is efficient in several special cases and mildly exponential in general. Finally, we prove that for identities with finite support, the map of the O'Hara's bijection can be computed in polynomial time, i.e. much more efficiently than by O'Hara's construction.Comment: 20 pages, 4 figure

Combinatorics and geometry of Littlewood-Richardson cones

We present several direct bijections between different combinatorial interpretations of the Littlewood-Richardson coefficients. The bijections are defined by explicit linear maps which have other applications.Comment: 15 pages, 9 figures. To be published in the special issue on "Combinatorics and Representation Theory" of the European Journal of Combinatoric

A Bijection between well-labelled positive paths and matchings

A well-labelled positive path of size n is a pair (p,\sigma) made of a word p=p_1p_2...p_{n-1} on the alphabet {-1, 0,+1} such that the sum of the letters of any prefix is non-negative, together with a permutation \sigma of {1,2,...,n} such that p_i=-1 implies \sigma(i)<\sigma(i+1), while p_i=1 implies \sigma(i)>\sigma(i+1). We establish a bijection between well-labelled positive paths of size $n$ and matchings (i.e. fixed-point free involutions) on {1,2,...,2n}. This proves that the number of well-labelled positive paths is (2n-1)!!. By specialising our bijection, we also prove that the number of permutations of size n such that each prefix has no more ascents than descents is [(n-1)!!]^2 if n is even and n!!(n-2)!! otherwise. Our result also prove combinatorially that the n-dimensional polytope consisting of all points (x_1,...,x_n) in [-1,1]^n such that the sum of the first j coordinates is non-negative for all j=1,2,...,n has volume (2n-1)!!/n!

The Method of Combinatorial Telescoping

We present a method for proving q-series identities by combinatorial telescoping, in the sense that one can transform a bijection or a classification of combinatorial objects into a telescoping relation. We shall illustrate this method by giving a combinatorial proof of Watson's identity which implies the Rogers-Ramanujan identities.Comment: 11 pages, 5 figures; to appear in J. Combin. Theory Ser.

A Baxter class of a different kind, and other bijective results using tableau sequences ending with a row shape

Tableau sequences of bounded height have been central to the analysis of k-noncrossing set partitions and matchings. We show here that familes of sequences that end with a row shape are particularly compelling and lead to some interesting connections. First, we prove that hesitating tableaux of height at most two ending with a row shape are counted by Baxter numbers. This permits us to define three new Baxter classes which, remarkably, do not obviously possess the antipodal symmetry of other known Baxter classes. We then conjecture that oscillating tableau of height bounded by k ending in a row are in bijection with Young tableaux of bounded height 2k. We prove this conjecture for k at most eight by a generating function analysis. Many of our proofs are analytic in nature, so there are intriguing combinatorial bijections to be found.Comment: 10 pages, extended abstrac

On the number of prime order subgroups of finite groups

Let G be a finite group and let ?(G) be the number of prime order subgroups of G. We determine the groups G with the property ?(G)??G?/2?1, extending earlier work of C. T. C. Wall, and we use our classification to obtain new results on the generation of near-rings by units of prime order