In 1998, the second author raised the problem of classifying the irreducible characters of S n of prime power degree. Zalesskii proposed the analogous problem for quasi-simple groups, and he has, in joint work with Malle, made substantial progress on this latter problem. With the exception of the alternating groups and their double covers, their work… (More)
Preface The history of quasisymmetric functions begins in 1972 with the thesis of Richard Stanley, followed by the formal definition of the Hopf algebra of quasisymmetric functions in 1984 by Ira Gessel. From this definition a whole research area grew and a more detailed, although not exhaustive, history can be found in the introduction. The history of… (More)
We present a general construction of involutions on integer partitions which enable us to prove a number of modulo 2 partition congru-ences.
Considering a question of E. R. Berlekamp, Carlitz, Roselle, and Scoville gave a combinatorial interpretation of the entries of certain matrices of determinant 1 in terms of lattice paths. Here we generalize this result by refining the matrix entries to be multivariate polynomials, and by determining not only the determinant but also the Smith normal form… (More)
— It is known that the two statistics on integer partitions " hook length " and " part length " are equidistributed over the set of all partitions of n. We extend this result by proving that the bivariate joint generating function by those two statistics is symmetric. Our method is based on a generating function by a triple statistic much easier to… (More)
The classical Littlewood-Richardson rule is a rule for computing coefficients in many areas, and comes in many guises. In this paper we prove two Littlewood-Richardson rules for symmetric skew quasisymmetric Schur functions that are analogous to the famed version of the classical Littlewood-Richardson rule involving Yamanouchi words. Furthermore, both our… (More)
Extending the partition function multiplicatively to a function on partitions, we show that it has a unique maximum at an explicitly given partition for any n = 7. The basis for this is an inequality for the partition function which seems not to have been noticed before.