Office: Building 4, Room 182 Phone: (617) 258-6895 Email: sassaf [AT] math [DOT] mit [DOT] edu Mailing Address:     Department of Mathematics     Massachusetts Institute of Technology     77 Massachusetts Avenue     Cambridge, MA 02139-4307 Curriculum Vita

(with Persi Diaconis and K. Soundararajan) A rule of thumb for riffle shuffling.

We study how many riffle shuffles are required to mix n cards if only certain features of the deck are of interest, e.g. suits disregarded or only the colors of interest. For a wide variety of features, the number of shuffles drops from 3/2 log_2 n to log_2 n. We derive closed formulae and an asymptotic `rule of thumb' formula which is remarkably accurate.
&bull  21 pages  &bull  preprint  &bull

A generalized major index statistic. Seminaire Lotharingien de Combinatoire 60 (2008), Art. B60c, 13 pp. (electronic).   arVix:0807.0433

Inspired by the k-inversion statistic for LLT polynomials, we define the k-inversion number and k-descent set for words. Using these, we define a new statistic on words, called the k-major index, that interpolates between the major index and inversion number. We give a bijective proof that the k-major index is equidistributed with the major index, generalizing a classical result of Foata and rediscovering a result of Kadell. Inspired by recent work of Haglund and Stevens, we give a partial extension of these definitions and constructions to standard Young tableaux. Finally, we give an application to Macdonald polynomials made possible by connections with LLT polynomials.

A combinatorial realization of Schur-Weyl duality via crystal graphs and dual equivalence graphs. Discrete Mathematics and Theoretical Computer Science, to appear as part of the conference proceedings for FPSAC 2008.   arVix:0804.1587
&bull  slides from MSRI Workshop (01/2008)  &bull

For any polynomial representation of the special linear group, the nodes of the corresponding crystal may be indexed by semi-standard Young tableaux. Under certain conditions, the standard Young tableaux occur, and do so with weight 0. Standard Young tableaux also parametrize the vertices of dual equivalence graphs. Motivated by the underlying representation theory, in this paper, we explain this connection by giving a combinatorial manifestation of Schur-Weyl duality. In particular, we put a dual equivalence graph structure on the 0-weight space of certain crystal graphs, producing edges combinatorially from the crystal edges. The construction can be expressed in terms of the local characterizations given by Stembridge for crystal graphs and the author for dual equivalence graphs.

The Schur expansion of Macdonald polynomials.

Building on Haglund's combinatorial formula for the transformed Macdonald polynomials, we provide a purely combinatorial proof of Macdonald positivity using dual equivalence graphs and give a combinatorial formula for the coefficients in the Schur expansion.
&bull  9 pages  &bull  preprint  &bull  slides from BIRS Workshop (09/2007)  &bull

A combinatorial proof of LLT and Macdonald positivity.

We make a systematic study of a new combinatorial construction called a dual equivalence graph. We axiomatize these graphs and prove that their generating functions are symmetric and Schur positive. By constructing a graph on ribbon tableaux which we transform into a dual equivalence graph, we give a combinatorial proof of the symmetry and Schur positivity of the ribbon tableaux generating functions introduced by Lascoux, Leclerc and Thibon. Using Haglund's formula for the transformed Macdonald polynomials, this also gives a combinatorial formula for the Schur expansion of Macdonald polynomials.
&bull 24 pages  &bull  preprint  &bull

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