Multivariate Juggling Probabilities

We consider refined versions of Markov chains related to juggling introduced by Warrington. We further generalize the construction to juggling with arbitrary heights as well as infinitely many balls, which are expressed more succinctly in terms of Markov chains on integer partitions. In all cases, we give explicit product formulas for the stationary probabilities. The normalization factor in one case can be explicitly written as a homogeneous symmetric polynomial. We also refine and generalize enriched Markov chains on set partitions. Lastly, we prove that in one case, the stationary distribution is attained in bounded time.Comment: 28 pages, 5 figures, final versio

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

Bumping sequences and multispecies juggling

Building on previous work by four of us (ABCN), we consider further generalizations of Warrington's juggling Markov chains. We first introduce "multispecies" juggling, which consist in having balls of different weights: when a ball is thrown it can possibly bump into a lighter ball that is then sent to a higher position, where it can in turn bump an even lighter ball, etc. We both study the case where the number of balls of each species is conserved and the case where the juggler sends back a ball of the species of its choice. In this latter case, we actually discuss three models: add-drop, annihilation and overwriting. The first two are generalisations of models presented in (ABCN) while the third one is new and its Markov chain has the ultra fast convergence property. We finally consider the case of several jugglers exchanging balls. In all models, we give explicit product formulas for the stationary probability and closed form expressions for the normalisation factor if known.Comment: 25 pages, 9 figures (v3: final version, several typos and figures fixed

Schubert calculus and shifting of interval positroid varieties

Consider k x n matrices with rank conditions placed on intervals of columns. The ranks that are actually achievable correspond naturally to upper triangular partial permutation matrices, and we call the corresponding subvarieties of Gr(k,n) the _interval positroid varieties_, as this class lies within the class of positroid varieties studied in [Knutson-Lam-Speyer]. It includes Schubert and opposite Schubert varieties, and their intersections, and is Grassmann dual to the projection varieties of [Billey-Coskun]. Vakil's "geometric Littlewood-Richardson rule" [Vakil] uses certain degenerations to positively compute the H^*-classes of Richardson varieties, each summand recorded as a (2+1)-dimensional "checker game". We use his same degenerations to positively compute the K_T-classes of interval positroid varieties, each summand recorded more succinctly as a 2-dimensional "K-IP pipe dream". In Vakil's restricted situation these IP pipe dreams biject very simply to the puzzles of [Knutson-Tao]. We relate Vakil's degenerations to Erd\H os-Ko-Rado shifting, and include results about computing "geometric shifts" of general T-invariant subvarieties of Grassmannians.Comment: 35 pp; this subsumes and obviates the unpublished http://arxiv.org/abs/1008.430