Falling stars: a fall-decorated rational shuffle theorem

2025/08/28 by Iraci, Alessandro, Pagaria, Roberto, Paolini, Giovanni
#05E05 #Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.2508.20935

Abstract

In this paper, we formulate a rational analog of the fall Delta theorem and the Delta square conjecture. We find a new dinv statistic on fall-decorated paths on a (m+k) × (n+k) rectangle that simultaneously extends the previously known dinv statistics on decorated square objects and non-decorated rectangular objects. We prove a symmetric function formula for the q,t-generating function of fall-decorated rectangular Dyck paths as a skewing operator applied to e_m,n+km and, conditionally on the rectangular paths conjecture, an analog formula for fall-decorated rectangular paths.

Citations

• A combinatorial skewing formula for the Rise Delta Theorem
• Delta and Theta Operator Expansions
• A proof of the Extended Delta Conjecture
• A Shuffle Theorem for Paths Under Any Line
• A proof of the compositional Delta conjecture
• The valley version of the Extended Delta Conjecture
• Poincare polynomials of character varieties, Macdonald polynomials and\n affine Springer fibers
• A proof of the Square Paths Conjecture
• The Delta Conjecture
• A proof of the shuffle conjecture
• Some remarkable new Plethystic Operators in the Theory of Macdonald Polynomials
• A simpler formula for the number of diagonal inversions of an\n (m,n)-Parking Function and a returning Fermionic formula
• Compositional (km,kn)-Shuffle Conjectures
• The Shuffle Algebra Revisited
• A computational and combinatorial exposé of plethystic calculus
• A compositional shuffle conjecture specifying touch points of the Dyck\n path
• A Combinatorial Formula for the Character of the Diagonal Coinvariants
• Hilbert schemes, polygraphs, and the Macdonald positivity conjecture

Related

• Decorated Dyck paths, the Delta conjecture, and a new q,t-square
• Bounce statistics for rational lattice paths
• Dinv, Area, and Bounce for k-Dyck paths
• Parallelogram polyominoes, partially labelled Dyck paths, and the Delta conjecture
• Symmetry of ascent and descent distributions on rectangular and staircase tableaux