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A combinatorial model for the Macdonald polynomials. (English) Zbl 1064.05147
Summary: We introduce a polynomial $$\overline{C}_\mu[Z;q,t]$$, depending on a set of variables $$Z= \{z_1, \dots\}$$, a partition $$\mu$$, and two extra parameters $$q$$, $$t$$. The definition of $$\overline{C}_\mu$$ involves a pair of statistics $$(\text{maj}(\sigma,\mu), \text{inv}(\sigma,\mu))$$ on words $$\sigma$$ of positive integers, and the coefficients of the $$z_i$$ are manifestly in $$\mathbb N[q,t]$$. We conjecture that $$\overline{C}_\mu[Z;q,t]$$ is none other than the modified Macdonald polynomial $$\overline{H}_\mu[Z; q, t]$$. We further introduce a general family of polynomials $$F_T[Z;q,S]$$, where $$T$$ is an arbitrary set of squares in the first quadrant of the $$xy$$ plane, and $$S$$ is an arbitrary subset of $$T$$. The coefficients of the $$F_T[Z;q,S]$$ are in $$\mathbb N[q]$$, and $$\overline{C}_\mu[Z;q,t]$$ is a sum of certain $$F_T[Z;q,S]$$ times nonnegative powers of $$t$$. We prove $$F_T[Z;q,S]$$ is symmetric in the $$z_i$$ and satisfies other properties consistent with the conjecture. We also show how the coefficient of a monomial in $$F_T[Z;q,S]$$ can be expressed recursively. MAPLE calculations indicate the $$F_T[Z;q,S]$$ are Schur-positive, and we present a combinatorial conjecture for their Schur coefficients when the set $$T$$ is a partition with at most three columns.

##### MSC:
 500000 Symmetric functions and generalizations
##### Keywords:
statistics; symmetric; Schur coefficients
Maple; SF
Full Text:
##### References:
 [1] Garsia, PNAS 90 (8) pp 3607– (1993) · Zbl 0831.05062 · doi:10.1073/pnas.90.8.3607 [2] J AM MATH SOC 14 pp 941– (2001) · Zbl 1009.14001 · doi:10.1090/S0894-0347-01-00373-3 [3] PROC AM MATH SOC 121 pp 469– (1994) [4] PROC AM MATH SOC 19 pp 236– (1968) · doi:10.1090/S0002-9939-1968-0223256-9 [5] MATH NACHR 83 pp 143– (1978) · Zbl 0319.05002 · doi:10.1002/mana.19780830111 [6] DISCRETE MATH 139 pp 219– (1995) · Zbl 0826.05056 · doi:10.1016/0012-365X(94)00134-5 [7] PROC AM MATH SOC 123 pp 2961– (1995) [8] ELECTRON J COMBIN 5 pp R45– (1998) [9] J SYMBOLIC COMPUT 20 pp 755– (1995) · Zbl 0849.68068 · doi:10.1006/jsco.1995.1077
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