The Pan–Skandera–Wang Map Is Bruhat-Increasing for Every n

Sihong Pan, Mark Skandera and Jiayuan Wang, Permanental Inequalities and Unit Interval Orders, EPTCS 445 (2026), arXiv:2606.13162v1, Section 7:

Conjecture 7.8. For all n and all w ∈ A_n we have w ≤ f_n(w) in the Bruhat order. A proof of Conjecture 7.8 would extend Theorem 5.1 to all totally nonnegative matrices.

Here A_n is the set of permutations whose first ⌊n/2⌋ entries permute {1, …, ⌊n/2⌋}, and f_n is defined by Algorithm 7.5 from f_4 through the insertion maps ins_p, inss_q and the reverse-complement conjugation. Theorem 7.7 of the paper states the case n ≤ 13, with proof omitted.

The frozen theorem D5/S3/Combinatorics/PanSkanderaWangBruhat.result proves the conjecture for every n ≥ 4, the range on which Algorithm 7.5 defines f_n. By Theorem 7.1 of the paper (Drake, Gerrish, Skandera), each comparison w ≤ f_n(w) makes the corresponding difference of monomials totally nonnegative, so the bijection gives the permanental inequality at h = ⌊n/2⌋ for all totally nonnegative matrices.

Issue 9706 records the screen made before the work. The arXiv record has only v1; the mathdb entry lists the conjecture as open without solution; google-deepmind/formal-conjectures and conjectures.io have no entry; searches by title, arXiv number and conjecture number found no later proof. Nothing under Problems/, D5/ or Library/, and no entry of the three screening records, concerns this paper. Mathlib has no Bruhat order on permutations, so the formal statement uses the tableau criterion of Björner and Brenti, Theorem 2.1.5.

Write R_x(p, q) = #{j ≤ p : x_j ≥ q} and, for 0 ≤ d ≤ min(p, n − p), S(p, d) = [p − d] ∪ {p − d + 2, p − d + 4, …, p + d}, a set of p positions. The invariant (F) for a pair (x, y) says R_x(p, q) ≤ #{j ∈ S(p, d) : y_j ≥ q} for all p, d, q; its case d = 0 is Bruhat domination. For w ∈ A_n let r be the position of n; in both parity branches of Algorithm 7.5 the insertion position of the output is s = 2r − n.

1. The base map f_4 satisfies (F) on its four inputs.
2. Reverse-complement conjugation preserves (F): the complement of the reflected selection S(p, d) is S(n − p, d), and R_{x^{RU}}(p, q) = p − q + 1 + R_x(n − p, n + 2 − q).
3. If (a, b) satisfies (F) and s = 2r − n, then (ins_r(a), inss_s(b)) satisfies (F). The proof compares the selected positions of the new word with those of the old one in six cases, by the position of s relative to p − d and p + d and by parity; only the last case uses s = 2r − n.
4. Deleting the maximum of w ∈ A_n gives an element of A_{n−1} for odd n and of Ã_{n−1} for even n; reverse-complementation maps Ã_{n−1} into A_{n−1}.

Induction on n gives (F) for every pair (w, f_n(w)), hence w ≤ f_n(w).

The statement would fail if some w ∈ A_n had R_w(p, q) > R_{f_n(w)}(p, q) for some p, q. It would fail to be the paper's statement if the insertion positions 2(p − k) − 1 and 2(q − k − 1) were read with a different k, or if the Bruhat order were replaced by a weaker order; the formal module transcribes Algorithm 7.5 literally, and its map was checked against an independent implementation on all of A_n for n ≤ 9.

Exhaustive checks of (F), its reverse-complement pairs, and every row of the insertion case table through n = 13 (4,251,652 source words), with an independent implementation agreeing through n = 9. The six-row insertion table was also checked by hand.

theorem; Tier 1 named open conjecture from a 2026 paper, preregistered in issue 9706 before the work. The computational use is none: the delivered statement is universally quantified over every n ≥ 4, and no declaration is a bounded enumeration, a checker, a numeric reduction or a certified instance.

The literature screen is bounded to the sources named above; citation indices were not exhaustively reachable, so no worldwide priority claim is made. The equivalence between the tableau criterion and the strong Bruhat order is taken from the literature and is not formalized here. The result settles Conjecture 7.8 at h = ⌊n/2⌋; the paper's further conjecture for all h is not addressed.