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International Mathematics Research Notices (2008) Vol. 2008 : article ID rnm157, 29 pages, doi:10.1093/imrn/rnm157 published on January 16, 2008
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© The Author 2008. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permissions@oxfordjournals.org

Nested Quantum Dyck Paths and {nabla}(s{lambda})

Nicholas A. Loehr1 and Gregory S. Warrington2,*

1 Department of Mathematics, Virginia Tech, Blacksburg, VA 24061-0123, USA
2 Department of Mathematics, Wake Forest University, Winston-Salem, NC 27109, USA

Correspondence: * Correspondence to be sent to: warrings{at}wfu.edu

We conjecture a combinatorial formula for the monomial expansion of the image of any Schur function under the Bergeron–Garsia nabla operator. The formula involves nested labeled Dyck paths weighted by area and a suitable "diagonal inversion" statistic. Our model includes as special cases many previous conjectures connecting thenabla operator to quantum lattice paths. The combinatorics of the inverse Kostka matrix leads to an elementary proof of our proposed formula when q = 1. We also outline a possible approach for proving all the extant nabla conjectures that reduces everything to the construction of sign-reversing involutions on explicit collections of signed, weighted objects.


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