LLT Polynomial

An LLT polynomial is an inversion-weighted generating function of tuples of semistandard skew tableaux (Lascoux et al. 1997). For a tuple , give a cell in component , row , and column shifted content . A pair of cells attacks when their shifted-content difference satisfies . An inversion is an attacking pair with , and

where the sum ranges over tuples whose rows weakly increase and columns strictly increase, and records all entries.

Flagged versions impose row-dependent upper bounds on tableau entries and need not be symmetric. For singleton diagrams of the same content, the standard flagged version is

Here is the number of pairs with . Alexandersson (2026) proved that a Demazure atom coefficient of contains , refuting the corresponding flagged atom-positivity conjecture. This concerns flagged atom expansions, not ordinary symmetric Schur positivity.