Journal of Combinatorics

Volume 3 (2012)

Number 3

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

Pages: 451 – 494



D. Armstrong (Department of Mathematics, University of Miami, Coral Gables, Florida, U.S.A.)

A. Garsia (Department of Mathematics, University of California at San Diego)

J. Haglund (Department of Mathematics, University of Pennsylvania, Philadelphia, Penn., U.S.A.)

B. Rhoades (Department of Mathematics, University of Southern California, Los Angeles, Calif., U.S.A.)

B. Sagan (Department of Mathematics, Michigan State University, East Lansing, Mich., U.S.A.)


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.


parking functions, diagonal harmonics, Tesler matrices

Published 19 February 2013