Let X=x_1,x_2,…,x_n and Y=y_1,y_2,…,y_n be sets of commuting variables, and let WPF_n denote the set of word parking functions of size n. For a word parking function PF, let area(PF), dinv(PF), X^PF, a_i(PF), d_i(PF), Rise(PF), and Val(PF) be the statistics and sets defined in the paper. Let Delta'_e_k denote the modified Delta operator associated with the elementary symmetric function e_k. Haglund–Remmel–Wilson's Delta Conjecture. For any integers n>k≥ 0, Delta'_e_ke_n=∑_PF∈WPF_nt^area(PF)q^dinv(PF)X^PF∏_i∈Rise(PF)(1+fraczt^a_i(PF))bigg|_z^n-k-1 =∑_PF∈WPF_nt^area(PF)q^dinv(PF)X^PF∏_i∈Val(PF)(1+fraczq^d_i(PF)+1)bigg|_z^n-k-1. This conjecture generalizes the Shuffle Theorem by giving two combinatorial formulas for the Delta-operator expression; the rise and valley sides are different generating functions on parking functions. The conjecture remains open in general, although several cases are known, including the case Delta'_e_1e_n, the rise version at q=1, the Catalan case, and cases where t or q is zero.

Haglund–Remmel–Wilson Delta Conjecture

From papers

Let $X=\{x_1,x_2,\ldots,x_n\}$ and $Y=\{y_1,y_2,\ldots,y_n\}$ be sets of commuting variables, and let $\mathcal{WPF}_n$ denote the set of word parking functions of size $n$ . For a word parking function $\mathrm{PF}$ , let $\mathrm{area}(\mathrm{PF})$ , $\mathrm{dinv}(\mathrm{PF})$ , $X^{\mathrm{PF}}$ , $a_i(\mathrm{PF})$ , $d_i(\mathrm{PF})$ , $\mathrm{Rise}(\mathrm{PF})$ , and $\mathrm{Val}(\mathrm{PF})$ be the statistics and sets defined in the paper. Let $\Delta'_{e_k}$ denote the modified Delta operator associated with the elementary symmetric function $e_k$ . Haglund–Remmel–Wilson's Delta Conjecture. For any integers $n>k\geq 0$ ,

\Delta'_{e_k}e_n=\sum_{\mathrm{PF}\in\mathcal{WPF}_n}t^{\mathrm{area}(\mathrm{PF})}q^{\mathrm{dinv}(\mathrm{PF})}X^{\mathrm{PF}}\prod_{i\in\mathrm{Rise}(\mathrm{PF})}\left(1+\frac{z}{t^{a_i(\mathrm{PF})}}\right)\bigg|_{z^{n-k-1}}

=\sum_{\mathrm{PF}\in\mathcal{WPF}_n}t^{\mathrm{area}(\mathrm{PF})}q^{\mathrm{dinv}(\mathrm{PF})}X^{\mathrm{PF}}\prod_{i\in\mathrm{Val}(\mathrm{PF})}\left(1+\frac{z}{q^{d_i(\mathrm{PF})+1}}\right)\bigg|_{z^{n-k-1}}.

This conjecture generalizes the Shuffle Theorem by giving two combinatorial formulas for the Delta-operator expression; the rise and valley sides are different generating functions on parking functions. The conjecture remains open in general, although several cases are known, including the case $\Delta'_{e_1}e_n$ , the rise version at $q=1$ , the Catalan case, and cases where $t$ or $q$ is zero.

Progress summary

Nothing recorded yet. Refresh searches the literature and the public web for attempts on this problem, and writes the first summary here.

Equivalent formulations 1

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

Haglund–Remmel–Wilson Delta Conjecture
Let $e_n$ be the elementary symmetric function, let $\Delta'_{e_{k-1}}$ be the modified Macdonald eigenoperator, and let $\mathcal{LD}_n$ be the set of labeled Dyck paths of size $n$ . For $P\in\mathcal{LD}_n$ , write $\operatorname{area}(P)$ and $\operatorname{dinv}(P)$ for its area and diagonal-inversion statistics, let $a_i(P)$ and $d_i(P)$ be the associated row statistics, let $\operatorname{Val}(P)$ be its set of contractible valleys, and let $\mathbf{x}^P$ denote its monomial weight. For positive integers $k\leq n$ , let $\{z^{n-k}\}$ extract the coefficient of $z^{n-k}$ . Haglund–Remmel–Wilson's Delta Conjecture.
$\Delta'_{e_{k-1}} e_n=\{z^{n-k}\}\left[\sum_{P\in\mathcal{LD}_n}q^{\operatorname{dinv}(P)}t^{\operatorname{area}(P)}\prod_{i\,:\,a_i(P)>a_{i-1}(P)}\left(1+z/t^{a_i(P)}\right)\mathbf{x}^P\right]$ $=\{z^{n-k}\}\left[\sum_{P\in\mathcal{LD}_n}q^{\operatorname{dinv}(P)}t^{\operatorname{area}(P)}\prod_{i\in\operatorname{Val}(P)}\left(1+z/q^{d_i(P)+1}\right)\mathbf{x}^P\right].$
This is a central conjectural combinatorial formula for the Delta operators, relating their symmetric-function expressions to weighted labeled Dyck paths. Its resolution status is not specified in the source.
source: Brendon Rhoades, “Generalizations of the flag variety tied to the Macdonald-theoretic delta operators”, arXiv:2204.03386 (2024).

Sources & referencesView supporting material

Primary source

Dun Qiu and Andrew Timothy Wilson, “The valley version of the Extended Delta Conjecture”, arXiv:1907.00268 (2019).

1 week ago