Combinatorial Gelfand models for some semigroups and q-rook monoid algebras

Inspired by the results of [R. Adin, A. Postnikov, Y. Roichman, Combinatorial Gelfand model, preprint math.RT arXiv:0709.3962], we propose combinatorial Gelfand models for semigroup algebras of some finite semigroups, which include the symmetric inverse semigroup, the dual symmetric inverse semigroup, the maximal factorizable subsemigroup in the dual symmetric inverse semigroup, and the factor power of the symmetric group. Furthermore we extend the Gelfand model for the semigroup algebras of the symmetric inverse semigroup to a Gelfand model for the $q$-rook monoid algebra.Comment: 14 page

The Algebra of Binary Search Trees

We introduce a monoid structure on the set of binary search trees, by a process very similar to the construction of the plactic monoid, the Robinson-Schensted insertion being replaced by the binary search tree insertion. This leads to a new construction of the algebra of Planar Binary Trees of Loday-Ronco, defining it in the same way as Non-Commutative Symmetric Functions and Free Symmetric Functions. We briefly explain how the main known properties of the Loday-Ronco algebra can be described and proved with this combinatorial point of view, and then discuss it from a representation theoretical point of view, which in turns leads to new combinatorial properties of binary trees.Comment: 49 page

Combinatorial Hopf algebra of superclass functions of type DD

We provide a Hopf algebra structure on the space of superclass functions on the unipotent upper triangular group of type D over a finite field based on a supercharacter theory constructed by Andr\'e and Neto. Also, we make further comments with respect to types B and C. Type A was explores by M. Aguiar et. al (2010), thus this paper is a contribution to understand combinatorially the supercharacter theory of the other classical Lie types.Comment: Last section modified. Recent development added and correction with respect to previous version state

Combinatorial operads from monoids

We introduce a functorial construction which, from a monoid, produces a set-operad. We obtain new (symmetric or not) operads as suboperads or quotients of the operads obtained from usual monoids such as the additive and multiplicative monoids of integers and cyclic monoids. They involve various familiar combinatorial objects: endofunctions, parking functions, packed words, permutations, planar rooted trees, trees with a fixed arity, Schr\"oder trees, Motzkin words, integer compositions, directed animals, and segmented integer compositions. We also recover some already known (symmetric or not) operads: the magmatic operad, the associative commutative operad, the diassociative operad, and the triassociative operad. We provide presentations by generators and relations of all constructed nonsymmetric operads.Comment: 42 pages. Complete version of the extended abstracts arXiv:1208.0920 and arXiv:1208.092