Publications and Preprints

Maximal projective degrees for strict partitions

co-authored with Anne Henke and Amitai Regev • Electronic Journal of Combinatorics Volume 14, Research Paper 59, August 2007, 15 pp. [PDF | PostScript]

Let λ be a partition, and denote by fλ the number of standard tableaux of shape λ. The asymptotic shape of λ maximizing fλ was determined in the classical work of Logan and Shepp and, independently, of Vershik and Kerov. The analogue problem, where the number of parts of λ is bounded by a fixed number, was done by Askey and Regev — though some steps in this work were assumed without a proof. Here these steps are proved rigorously. When λ is strict, we denote by gλ the number of standard tableau of shifted shape λ. We determine the partition λ maximizing gλ in the strip. In addition we give a conjecture related to the maximizing of gλ without any length restrictions.

Strict Partitions of Maximal Projective Degree

Preprint arXiv:0705.4015v1 [PDF | PostScript].

The projective degrees of strict partitions of n were computed for all n ≤ 100 and the partitions with maximal projective degree were found for each n. It was observed that maximizing partitions for successive values of n “lie close to each other” in a certain sense. Conjecturing that this holds for larger values of n, the partitions of maximal degree were computed for all n ≤ 220. The results are consistent with a recent conjecture on the limiting shape of the strict partition of maximal projective degree.

A maj-inv bijection for C2≀An

Preprint arXiv:math.CO/0509239 [PDF | PostScript].

We give a bijective proof of the MacMahon-type equidistribution over the group of signed even permutations C2≀An that was stated in [2]. This is done by generalizing the bijection that was introduced in the bijective proof of the equidistribution over the alternating group An in [1].

A Foata bijection for the alternating group and for q-analogues

co-authored with Amitai Regev • Séminaire Lotharingien de Combinatoire, Volume 53, 2005, Art. B53b, 16pp. [PDF | PostScript].

The Foata bijection Φ:Sn→Sn is extended to the bijections Ψ:An+1→An+1 and Ψq:Sn+q−1→Sn+q−1, where Sm, Am are the symmetric and the alternating groups. These bijections imply bijective proofs for recent equidistribution theorems, by Regev and Roichman, for An+1 and for Sn+q−1.

A combinatorial proof of a bibasic trigonometric identity

European Journal of Combinatorics Volume 27, Issue 4, May 2006, Pages 518–525 doi:10.1016/j.ejc.2005.01.007

The bibasic trigonometric functions, recently introduced by Foata and Han, give rise to the p,q-tangent numbers and the p,q-secant numbers. Foata and Han proposed a combinatorial interpretation of these bibasic coefficients as enumerations of alternating permutations by the bi-statistic (inv1,inv2). Under this interpretation, the symmetry of the bibasic trigonometric functions yields a combinatorial identity. A combinatorial proof of the identity is desired. For permutations of even order, this has already been given by Foata and Han. Here we give a proof for permutations of odd order.

Euler-Mahonian polynomials for Ca≀Sn

Advances in Applied Mathematics Volume 37, Issue 2, August 2006, Pages 198–208 doi:10.1016/j.aam.2005.09.004

The L-reverse major index statistic, rmajL, is defined on the group of colored permutations, Ca≀Sn, using the <L order and the L-descent number statistic, desL, which were recently introduced by Regev and Roichman. The distribution of desL and the bi-statistic (desL, rmajL) is studied, yielding new wreath-product analogues of the Eulerian and q-Euler-Mahonian polynomials, and a generalization of Carlitz’s identity.

Earlier version available as arXiv:math.CO/0412091.

MacMahon-type Identities for Signed Even Permutations

Electronic Journal of Combinatorics Volume 11, Research Paper 83, November 2004, 18 pp. [PDF | PostScript]

MacMahon’s classic theorem states that the length and major index statistics are equidistributed on the symmetric group Sn. By defining natural analogues or generalizations of those statistics, similar equidistribution results have been obtained for the alternating group An by Regev and Roichman, for the hyperoctahedral group Bn by Adin, Brenti and Roichman, and for the group of even-signed permutations Dn by Biagioli. We prove analogues of MacMahon’s equidistribution theorem for the group of signed even permutations and for its subgroup of even-signed even permutations.

The computational complexity of rules for the character table of Sn

Journal of Symbolic Computation Volume 37, Issue 6, June 2004, Pages 727–748 doi:10.1016/j.jsc.2003.11.001

The Murnaghan-Nakayama rule is the classical formula for computing the character table of Sn. Y. Roichman has recently discovered a rule for the Kazhdan-Lusztig characters of q-Hecke algebras of type A, which can also be used for the character table of Sn. For each of the two rules, we give an algorithm for computing entries in the character table of Sn. We then analyze the computational complexity of the two algorithms, and in the case of characters indexed by partitions in the (k,l)-hook, compare their complexities to each other. It turns out that the algorithm based on the Murnaghan-Nakayama rule requires far less operations than the other algorithm. We note the algorithms’ complexities’ relation to two enumeration problems of Young diagrams and Young tableaux.

Earlier version available as arXiv:math.CO/0309225.