# Sign and maj imbalance of posets

## ABSTRACT

Let P be a partial ordering of 1,2,...,n. The sign imbalance ofP is defined by I_P = \sum_\pi (-1)^{inv(\pi)}, where \pi ranges over all linear extensions of P (regarded as permutations of 1,2,...,n) and inv(\pi) denotes the number of inversions of \pi. The maj imbalance is defined analogously with inv replaced by maj (the major index). We will give a survey of sign and maj imbalance, including connections with promotion and evacuation, domino tilings, P-partitions, and symmetric functions.

Fairly easy exercise: If P is a product of two chains of even cardinalities, then I_P=0.

Much more difficult exercise: If P is a product of two chains of odd cardinalities >1, then I_P=0.

Speaker's Contact Info: rstan(at-sign)math.mit.edu