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{\bf Mike Zabrocki}
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{\bf A Macdonald Vertex Operator and Standard Tableaux Statistics}
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The two parameter family of coefficients $K_{\lambda \mu}(q,t)$
introduced by Macdonald are conjectured to $(q,t)$ count the standard
tableaux of shape $\lambda $.  If this conjecture is correct, then there
exist statistics $a_\mu(T)$ and $b_\mu(T)$ such that the family of
symmetric functions $H_\mu[X;q,t] = \sum_\lambda  K_{\lambda  \mu}(q,t) s_\lambda [X]$
are generating functions for the
standard tableaux of size $|\mu|$ in the sense that
$H_\mu[X;q,t] = \sum_{T} q^{a_\mu(T)} t^{b_\mu(T)} s_{\lambda (T)}[X]$
where the sum is over standard
tableau of of size $|\mu|$.  We give a formula for a symmetric function
operator $H_2^{qt}$ with the property that $H_2^{qt} H_{(2^a1^b)}[X;q,t]=
H_{(2^{a+1}1^b)}[X;q,t]$.  This operator has a combinatorial action
on the Schur function basis.  We use this Schur function
action to show by induction that
$H_{(2^a1^b)}[X;q,t]$ is the generating function for standard tableaux
of size $2a+b$ (and hence that $K_{\lambda (2^a1^b)}(q,t)$ is a
polynomial with non-negative integer coefficients).
The inductive proof gives an algorithm for 'building' the
standard tableaux of size $n+2$  from the
standard tableaux of size $n$ and divides the standard tableaux into
classes that are generalizations of the catabolism type.
We show that reversing this construction gives the
statistics $a_\mu(T)$ and $b_\mu(T)$ when $\mu$ is of the form
$(2^a1^b)$ and that these statistics prove conjectures about
the relationship between adjacent rows of the $(q,t)$-Kostka
matrix that were suggested by Lynne Butler.


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