Generalized catalan numbers, sequences and polynomials

In this paper we present an algebraic interpretation for generalized Catalan numbers. We describe them as dimensions of certain subspaces of multilinear polynomials. This description is of utmost importance in the investigation of annihilators in exterior algebras. | Turk J Math 34 (2010) , 441 – 449. ¨ ITAK ˙ c TUB doi: Generalized catalan numbers, sequences and polynomials ˙ Cemal Ko¸c, Ismail G¨ ulo˘glu, Song¨ ul Esin Abstract In this paper we present an algebraic interpretation for generalized Catalan numbers. We describe them as dimensions of certain subspaces of multilinear polynomials. This description is of utmost importance in the investigation of annihilators in exterior algebras. 1. Introduction Let V be a vector space over a field F and X ⊆ V. An element μ = ς1 + ς2 + . . . + ςn of the exterior algebra E(V ) of V is said to be neat with respect to X if ςi = xi1 ∧xi2 ∧. . .∧xini with xij ∈ X , j = 1, 2, . . . , ni and ς1 ∧ ς2 ∧ . . . ∧ ςr = 0. The annihilator of μ in E(V ) is described by products of the form (ςi1 − ςj1 ) . . . (ςir − ςjr )ςk1 . . . ςkt when Char(F ) = 0 (see [1]). Dimensions of subspaces of E(V ) spanned by certain elements of this type can be used to extend results of [1] to remove the restriction Char(F ) = 0. Motivated by this, we continue the study of certain type of ideals of the polynomial ring, studied in [2]. To be more precise, let F [z] = F [z1 , . . . , zn ] be the ring of polynomials in n indeterminates over F . The symmetric group Sn of degree n acts on this ring canonically as f σ (z) = f(zσ(1) , . . . , zσ(n) ). Letting p(z) = (z1 − z2 ) . . . (z2r−1 − z2r ) (z1 − z2 ) . . . (z2r−1 − z2r )z2r+1 if n = 2r if n = 2r + 1, we can form the cyclic module F [Sn ]p(z) over the group ring F [Sn ]. In [2] it was proved that F [Sn ]p(z) = F [H]p(z), 2000 AMS Mathematics Subject Classification: . 441 ¨ ˘ ˙ KOC ¸ , GULO GLU, ESIN where H is the subgroup of Sn fixing each z2k , k = 1, . . . , r. Identifying H with Sn−r and setting z2i−1 = xi for i = 1, . . . , n − r , z2j = yj for j = 1, . . . , r, F [z] = pσ (z) = F [x; y] and p(z) = p(x; y) pσ (x; y) = p(xσ(1) , . . . , xσ(n−r) ; y1 , . . . , yr ) for σ ∈ Sn−r the results given in Theorem 6 and its .