The number of monotone triangles with prescribed bottom row

We show that the number of monotone triangles with prescribed bottom row (k_1,...,k_n) is given by a simple product formula which remarkably involves (shift) operators. Monotone triangles with bottom row (1,2,...,n) are in bijection with $n \times n$ alternating sign matrices

A symmetry theorem on a modified jeu de taquin

For their bijective proof of the hook-length formula for the number of standard tableaux of a fixed shape Novelli, Pak and Stoyanovskii define a modified jeu de taquin which transforms an arbitrary filling of the Ferrers diagram with $1,2,...,n$ (tabloid) into a standard tableau. Their definition relies on a total order of the cells in the Ferrers diagram induced by a special standard tableau, however, this definition also makes sense for the total order induced by any other standard tableau. Given two standard tableaux $P,Q$ of the same shape we show that the number of tabloids which result in $P$ if we perform modified jeu de taquin with respect to the total order induced by $Q$ is equal to the number of tabloids which result in $Q$ if we perform modified jeu de taquin with respect to $P$. This symmetry theorem extends to skew shapes and shifted skew shapes.Comment: 8 page

Process nominalizations in russian

Within a minimalist framework of sound-meaning correlation, the present study concentrates on process nominalizations of Russian. It is shown how these constructions are built up syntactically and semantically and in which respects they differ from other types of nominalizations. The analysis follows a lexicalist conception of word formation and the differentiation of Semantic Form and Conceptual Structure