{TALK 
        {"December 4}
        {"Sign and maj imbalance of posets}
        {"Richard Stanley}
        {"MIT}
        {"rstan@math.mit.edu}
        {"
Let <em>P</em> be a partial ordering of 1,2,...,<em>n</em>. The
<em>sign imbalance</em> of<em>P</em> is defined by <em>I_P</em> =
\sum_\pi (-1)^{inv(\pi)}, where \pi ranges over all linear extensions
of <em>P</em> (regarded as permutations of 1,2,...,<em>n</em>) and
inv(\pi) denotes the number of inversions of \pi. The <em>maj
imbalance</em> 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,
<em>P</em>-partitions, and symmetric functions.
<p>
Fairly easy exercise: If <em>P</em> is a product of two chains of even
cardinalities, then <em>I_P</em>=0.
<p>
Much more difficult exercise: If <em>P</em> is a product of two chains
of odd cardinalities >1, then <em>I_P</em>=0.
}
}