A weighted sum over generalized Tesler matrices

We generalize previous definitions of Tesler matrices to allow negative matrix entries and negative hook sums. Our main result is an algebraic interpretation of a certain weighted sum over these matrices, which we call the Tesler function. Our interpretation uses a new class of symmetric function specializations which are defined by their values on Macdonald polynomials. As a result of this interpretation, we obtain a Tesler function expression for the Hall inner product $\langle Δ_f e_n, p_{1^{n}}\rangle$, where $Δ_f$ is the delta operator introduced by Bergeron, Garsia, Haiman, and Tesler. We also provide simple formulas for various special cases of Tesler functions which involve $q,t$-binomial coefficients, ordered set partitions, and parking functions. These formulas prove two cases of the recent Delta Conjecture posed by Haglund, Remmel, and the author.

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