Square $\boldsymbol {q,t}$-lattice paths and $\boldsymbol {\nabla (p_n)}$

DOI10.1090/S0002-9947-06-04044-XzbMath1107.05098WikidataQ114093928 ScholiaQ114093928MaRDI QIDQ3420285

Gregory S. Warrington, Nicholas A. Loehr

Publication date: 1 February 2007

Published in: Transactions of the American Mathematical Society (Search for Journal in Brave)

zbMATH Keywords

lattice pathsMacdonald polynomialsDyck pathsCatalan numbersdiagonal harmonicsnabla operator

Mathematics Subject Classification ID

(q)-calculus and related topics (05A30) Combinatorial aspects of representation theory (05E10) Representations of finite symmetric groups (20C30)

Related Items (16)

A combinatorial expansion of vertical-strip LLT polynomials in the basis of elementary symmetric functions ⋮ A proof of the \(q,t\)-square conjecture ⋮ A proof of the square paths conjecture ⋮ A valley version of the delta square conjecture ⋮ Decorated Dyck Paths, Polyominoes, and the Delta Conjecture ⋮ Limits of modified higher \(q,t\)-Catalan numbers ⋮ Combinatorics of the Delta conjecture at \(q=-1\) ⋮ Rectangular analogues of the square paths conjecture and the univariate Delta conjecture ⋮ The Delta Conjecture ⋮ Theta operators, refined delta conjectures, and coinvariants ⋮ New invariants for permutations, orders and graphs ⋮ Sweep maps: a continuous family of sorting algorithms ⋮ The generalized Delta conjecture at \(t = 0\) ⋮ The shuffle conjecture ⋮ Unnamed Item ⋮ Delta conjectures

Cites Work

• Lattice diagram polynomials and extended Pieri rules
• Vanishing theorems and character formulas for the Hilbert scheme of points in the plane
• Identities and positivity conjectures for some remarkable operators in the theory of symmetric functions
• A combinatorial formula for the character of the diagonal coinvariants
• Conjectured combinatorial models for the Hilbert series of generalized diagonal harmonic modules
• Conjectured statistics for the higher \(q,t\)-Catalan sequences
• Combinatorics of \(q\), \(t\)-parking functions
• A proof of the \(q,t\)-Catalan positivity conjecture
• A Schröder generalization of Haglund's statistic on Catalan paths
• Conjectured statistics for the \(q,t\)-Catalan numbers.
• A remarkable \(q,t\)-Catalan sequence and \(q\)-Lagrange inversion
• Trapezoidal lattice paths and multivariate analogues
• A conjectured combinatorial formula for the Hilbert series for diagonal harmonics
• A positivity result in the theory of Macdonald polynomials
• Hilbert schemes, polygraphs and the Macdonald positivity conjecture
• Combinatorial theory of Macdonald polynomials I: Proof of Haglund's formula
• A combinatorial model for the Macdonald polynomials
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