17 research outputs found
Growth diagrams, Domino insertion and Sign-imbalance
We study some properties of domino insertion, focusing on aspects related to
Fomin's growth diagrams. We give a self-contained proof of the semistandard
domino-Schensted correspondence given by Shimozono and White, bypassing the
connections with mixed insertion entirely. The correspondence is extended to
the case of a nonempty 2-core and we give two dual domino-Schensted
correspondences. We use our results to settle Stanley's `2^{n/2}' conjecture on
sign-imbalance and to generalise the domino generating series of Kirillov,
Lascoux, Leclerc and Thibon.Comment: 24 page
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
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
Some remarks on sign-balanced and maj-balanced posets
Let P be a poset with elements 1,2,...,n. We say that P is sign-balanced if
exactly half the linear extensions of P (regarded as permutations of 1,2,...,n)
are even permutations, i.e., have an even number of inversions. This concept
first arose in the work of Frank Ruskey, who was interested in the efficient
generation of all linear extensions of P. We survey a number of techniques for
showing that posets are sign-balanced, and more generally, computing their
"imbalance." There are close connections with domino tilings and, for certain
posets, a "domino generalization" of Schur functions due to Carre and Leclerc.
We also say that P is maj-balanced if exactly half the linear extensions of P
have even major index. We discuss some similarities and some differences
between sign-balanced and maj-balanced posets.Comment: 30 pages. Some inaccuracies in Section 3 have been corrected, and
Conjecture 3.6 has been adde
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