#####
Séminaire Lotharingien de Combinatoire, B54As (2007), 55 pp.

# Jeffrey Remmel

# The Combinatorics of Macdonald's *D*_{n}^{1} Operator

**Abstract.**
To prove the existence of the Macdonald polynomials
{*P*_{\la}(*x*;*q*,*t*)}, *\la* a partition of *n*,
Macdonald
[*Séminaire Lotharingien Combin.* **20** (1988), Article B20a;
*"Symmetric functions and Hall polynomials"*,
2nd ed., Clarendon Press, New York, 1995]
introduced an operator *D*_{n}^{1} and proved that for any
Schur function *s*_{\la}(*x*_{1}, ..., *x*_{n}),
*D*_{n}^{1} *s*_{\la}(*x*_{1}, ..., *x*_{n}) =
*\sum*_{\mu} d_{\la,\mu}(*q*,*t*)
*s*_{\mu}(*x*_{1}, ..., *x*_{n})
where the sum runs over all partitions *\mu* of *n* which are
less than or equal to *\la* in the dominance order and the
*d*_{\la,\mu}(*q*,*t*) are polynomials in *q* and *t*
with integer coefficients. We give an explicit combinatorial formula for the *d*_{\la,\mu}(*q*,*t*)'s.

Received: December 11, 2005.
Accepted: September 14, 2006.
Final Version: January 5, 2007.

The following versions are available: