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In (J. Haglund, A polynomial expression for the Hilbert series of the quotient ring of diagonal coinvariants, Adv. Math. 227 (2011) 2092-2106), the study of the Hilbert series of diagonal coinvariants is linked to combinatorial objects called Tesler matrices. In this paper we use operator identities from Macdonald polynomial theory to give new and short proofs of some of these results. We also develop the combinatorial theory of Tesler matrices and parking functions, extending results of P. Levande, and apply our results to prove various special cases of a positivity conjecture of Haglund.

Combinatorics of Tesler matrices in the theory of parking functions and diagonal harmonics