23 research outputs found
A geometric version of the Robinson–Schensted correspondence for skew oscillating tableaux
AbstractWe consider an analogue of the Robinson–Schensted correspondence for skew oscillating tableaux and we propose a geometric version of this correspondence, extending similar constructions for standard (Combinatoire et représentation du groupe symétrique, Lecture Notes in Mathematics, Vol. 579, Springer, Berlin, 1977, pp. 29–58) and oscillating tableaux (Formal Power Series and Algebraic Combinatorics, FPSAC’99, Univ. Politecnicà de Catalunya, 1999, pp. 141–152). We deduce from this geometric construction new proofs of some combinatorial properties of this correspondence
Descent sets for symplectic groups
The descent set of an oscillating (or up-down) tableau is introduced. This
descent set plays the same role in the representation theory of the symplectic
groups as the descent set of a standard tableau plays in the representation
theory of the general linear groups. In particular, we show that the descent
set is preserved by Sundaram's correspondence. This gives a direct
combinatorial interpretation of the branching rules for the defining
representations of the symplectic groups; equivalently, for the Frobenius
character of the action of a symmetric group on an isotypic subspace in a
tensor power of the defining representation of a symplectic group.Comment: 22 pages, 2 figure
Growth diagrams, and increasing and decreasing chains in fillings of Ferrers shapes
We put recent results by Chen, Deng, Du, Stanley and Yan on crossings and
nestings of matchings and set partitions in the larger context of the
enumeration of fillings of Ferrers shape on which one imposes restrictions on
their increasing and decreasing chains. While Chen et al. work with
Robinson-Schensted-like insertion/deletion algorithms, we use the growth
diagram construction of Fomin to obtain our results. We extend the results by
Chen et al., which, in the language of fillings, are results about
--fillings, to arbitrary fillings. Finally, we point out that, very
likely, these results are part of a bigger picture which also includes recent
results of Jonsson on --fillings of stack polyominoes, and of results of
Backelin, West and Xin and of Bousquet-M\'elou and Steingr\'\i msson on the
enumeration of permutations and involutions with restricted patterns. In
particular, we show that our growth diagram bijections do in fact provide
alternative proofs of the results by Backelin, West and Xin and by
Bousquet-M\'elou and Steingr\'\i msson.Comment: AmS-LaTeX; 27 pages; many corrections and improvements of
short-comings; thanks to comments by Mireille Bousquet-Melou and Jakob
Jonsson, the final section is now much more profound and has additional
result
Stammering tableaux
The PASEP (Partially Asymmetric Simple Exclusion Process) is a probabilistic
model of moving particles, which is of great interest in combinatorics, since
it appeared that its partition function counts some tableaux. These tableaux
have several variants such as permutations tableaux, alternative tableaux,
tree- like tableaux, Dyck tableaux, etc. We introduce in this context certain
excursions in Young's lattice, that we call stammering tableaux (by analogy
with oscillating tableaux, vacillating tableaux, hesitating tableaux). Some
natural bijections make a link with rook placements in a double staircase,
chains of Dyck paths obtained by successive addition of ribbons, Laguerre
histories, Dyck tableaux, etc.Comment: Clarification and better exposition thanks reviewer's report