Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2506.23373

The monomial expansions of modified Macdonald polynomials

Emma Yu Jin·Xiaowei Lin
View original source ↗Read source document ↗Export metadata ↓

Abstract

We discover a family $A$ of sixteen statistics on fillings of any given Young diagram and prove new combinatorial formulas for modified Macdonald polynomials, that is, $$\tilde{H}_λ(X;q,t)=\sum_{σ\in T(λ)}x^σq^{maj(σ)}t^{η(σ)}$$ for each statistic $η\in A$. Building upon this new formula, we establish four compact formulas for the modified Macdonald polynomials, namely, $$\tilde{H}_λ(X;q,t)=\sum_σd_{\varepsilon}(σ)x^σq^{maj(σ)}t^{η(σ)}$$ which is summed over all canonical or dual canonical fillings of a Young diagram and $d_{\varepsilon}(σ)$ is a product of $t$-multinomials. Finally, the compact formulas enable us to derive four explicit expressions for the monomial expansion of modified Macdonald polynomials, one of which coincides with the formula given by Garbali and Wheeler (2020).

Explore related subjects

math.COmath-phmath.MP

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Emma Yu Jin, Xiaowei Lin. 2025-07-05. The monomial expansions of modified Macdonald polynomials. https://arxiv.org/abs/2506.23373

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

On nut graphs with two vertex and three edge orbits

Nut graphs are graphs whose adjacency matrix is singular with one-dimensional null space spanned by a vector with no zero entries. In a recent paper, Bašić, Fowler and Pisanski proved that the automorphism group of a nut graph has more orbits on the edge set than on the vertex set. They classified all orders for which a vertex-transitive nut graph with precisely two edge orbits exists, and conjectured that a nut graph with two vertex and three edge orbits exists for each non-prime order $n \ge 9$. Motivated by this conjecture, we introduce a very general construction that provides graphs with the desired symmetry properties, and we determine some sufficient spectral and structural conditions under which they are nut graphs. The construction yields infinite families of examples and confirms the above conjecture for all odd non-prime orders up to $2\,500$ and for at least $99.8$ percent of all odd non-prime orders up to a million. Finally, we present some additional interesting examples of nut graphs with two vertex and three edge orbits that do not arise from this construction.

math.CO↗
arXiv2026

On vertex-minimal simplicial maps to the sphere

For positive integers $n,d$, let $λ(n,d)$ be the minimal number of vertices of a triangulation of the $n$-sphere which admits a degree $d$ simplicial map onto the boundary of the $(n+1)$-simplex. We show that for $h=\lfloor\frac{n+1}2\rfloor$, the function $λ(n,d)^h$ has linear order of growth in $d$, answering a question of O. Musin. All triangulations we obtained are isomorphic to boundaries of convex polytopes in $\mathbb{R}^{n+1}$.

math.CO↗