Wide partitions, Latin tableaux, and Rota's basis conjecture

Say that mu is a ``subpartition'' of an integer partition lambda if the multiset of parts of mu is a submultiset of the parts of lambda, and define an integer partition lambda to be ``wide'' if for every subpartition mu of lambda, mu >= mu' in dominance order (where mu' denotes the conjugate or transpose of mu). Then Brian Taylor and the first author have conjectured that an integer partition lambda is wide if and only if there exists a tableau of shape lambda such that (1) for all i, the entries in the ith row of the tableau are precisely the integers from 1 to lambda_i inclusive, and (2) for all j, the entries in the jth column of the tableau are pairwise distinct. This conjecture was originally motivated by Rota's basis conjecture and, if true, yields a new class of integer multiflow problems that satisfy max-flow min-cut and integrality. Wide partitions also yield a class of graphs that satisfy ``delta-conjugacy'' (in the sense of Greene and Kleitman), and the above conjecture implies that these graphs furthermore have a completely saturated stable set partition. We present several partial results, but the conjecture remains very much open.Comment: Joined forces with Goemans and Vondrak---several new partial results; 28 pages, submitted to Adv. Appl. Mat

Han's Bijection via Permutation Codes

We show that Han's bijection when restricted to permutations can be carried out in terms of the cyclic major code and the cyclic inversion code. In other words, it maps a permutation $\pi$ with a cyclic major code $(s_1, s_2, ..., s_n)$ to a permutation $\sigma$ with a cyclic inversion code $(s_1,s_2, ..., s_n)$. We also show that the fixed points of Han's map can be characterized by the strong fixed points of Foata's second fundamental transformation. The notion of strong fixed points is related to partial Foata maps introduced by Bj\"orner and Wachs.Comment: 12 pages, to appear in European J. Combi

Large magnetothermal conductivity of HoMnO_3 single crystals and its relation to the magnetic-field induced transitions of magnetic structure

We study the low-temperature heat transport of HoMnO_3 single crystals to probe the magnetic structures and their transitions induced by magnetic field. It is found that the low-T thermal conductivity (\kappa) shows very strong magnetic-field dependence, with the strongest suppression of nearly 90% and the biggest increase of 20 times of \kappa compared to its zero-field value. In particular, some ``dip"-like features show up in \kappa(H) isotherms for field along both the ab plane and the c axis. These behaviors are found to shed new light on the complex H-T phase diagram and the field-induced re-orientations of Mn^{3+} and Ho^{3+} spin structures. The results also demonstrate a significant spin-phonon coupling in this multiferroic compound.Comment: 5 pages, 4 figures, accepted for publication in Phys. Rev.