Deterministic and Probabilistic Bijective Combinatorics for Macdonald Polynomials

Abstract

Permuted-basement Macdonald polynomials 𝐸^𝜎_𝛼(𝐱; 𝑞, 𝑡) are nonsymmetric generalizations of symmetric Macdonald polynomials indexed by a composition 𝛼 and a permutation 𝜎. They form a basis for the polynomial ring ℚ(𝑞, 𝑡)[𝐱] for each fixed permutation 𝜎. They can be described combinatorially as generating functions over augmented fillings of composition shape 𝛼 with a basement permutation 𝜎.

We construct deterministic bijections and probabilistic bijections on fillings that prove identities relating 𝐸^𝜎_𝛼, 𝐸^{𝜎𝑠ᵢ}_𝛼, 𝐸^𝜎_{𝑠ᵢ𝛼}, and 𝐸^{𝜎𝑠ᵢ}_{𝑠ᵢ𝛼}. These identities correspond to two combinatorial operations on the shape and basement of the fillings: swapping adjacent parts in the shape, which expands 𝐸^𝜎_𝛼 in terms of 𝐸^𝜎_{𝑠ᵢ𝛼} and 𝐸^{𝜎𝑠ᵢ}_{𝑠ᵢ𝛼}; and swapping adjacent entries in the basement, which gives 𝐸^𝜎_𝛼 = 𝐸^{𝜎𝑠ᵢ}_𝛼 when 𝛼ᵢ = 𝛼ᵢ₊₁.