On the sign-imbalance of partition shapes

Let the sign of a standard Young tableau be the sign of the permutation you get by reading it row by row from left to right, like a book. A conjecture by Richard Stanley says that the sum of the signs of all SYTs with n squares is 2^[n/2]. We present a stronger theorem with a purely combinatorial proof using the Robinson-Schensted correspondence and a new concept called chess tableaux. We also prove a sharpening of another conjecture by Stanley concerning weighted sums of squares of sign-imbalances. The proof is built on a remarkably simple relation between the sign of a permutation and the signs of its RS-corresponding tableaux.Comment: 12 pages. Better presentatio

On the sign-imbalance of skew partition shapes

Let the sign of a skew standard Young tableau be the sign of the permutation you get by reading it row by row from left to right, like a book. We examine how the sign property is transferred by the skew Robinson-Schensted correspondence invented by Sagan and Stanley. The result is a remarkably simple generalization of the ordinary non-skew formula. The sum of the signs of all standard tableaux on a given skew shape is the sign-imbalance of that shape. We generalize previous results on the sign-imbalance of ordinary partition shapes to skew ones.Comment: 14 pages; former section 8 is removed and the rest is slightly update

The Hopf algebra of odd symmetric functions

We consider a q-analogue of the standard bilinear form on the commutative ring of symmetric functions. The q=-1 case leads to a Z-graded Hopf superalgebra which we call the algebra of odd symmetric functions. In the odd setting we describe counterparts of the elementary and complete symmetric functions, power sums, Schur functions, and combinatorial interpretations of associated change of basis relations.Comment: 43 pages, 12 figures. v2: some correction

Enumeration of Standard Young Tableaux

A survey paper, to appear as a chapter in a forthcoming Handbook on Enumeration.Comment: 65 pages, small correction

Odd symmetric functions and categorification

We introduce q- and signed analogues of several constructions in and around the theory of symmetric functions. The most basic of these is the Hopf superalgebra of odd symmetric functions. This algebra is neither (super-)commutative nor (super-)cocommutative, yet its combinatorics still exhibit many of the striking integrality and positivity properties of the usual symmetric functions. In particular, we give odd analogues of Schur functions, Kostka numbers, and Littlewood-Richardson coefficients. Using an odd analogue of the nilHecke algebra, we give a categorification of the integral divided powers form of U_q^+(sl_2) inequivalent to the one due to Khovanov-Lauda. Along the way, we develop a graphical calculus for indecomposable modules for the odd nilHecke algebra