Palindromicity of shifted lower Bruhat intervals: the (w,x) census for n ≤ 6, the 3412/4231 separation, and the failure of the Carrell–Peterson dictionary

Abstract

For elements w,x of a Coxeter group W, the shifted lower Bruhat interval is the right translate S(w,x)={vx⁻¹:v ≤ w} of the principal order ideal of w, carrying the order induced from the ambient Bruhat order. Dave, Feng, Lesnevich, Oh, Richmond, Wang and Wu introduced it and proved that it has a unique maximum and a unique minimum, that its covering relations are ambient, that it is graded, and that it is EL-shellable. Gradedness makes its rank generating function F_(w,x)(q) well defined, and at x=e it is the Poincaré polynomial of the Schubert variety X_w, palindromic exactly when X_w is rationally smooth. We ask for which pairs (w,x) the polynomial F_(w,x) is palindromic — a question the source does not raise — and settle a first round of it in type A. We give the complete census in Sₙ for n ≤ 6: the palindromic pairs number 4, 36, 464, 8228, 171112; the permutations palindromic for every shift number 2,6,15,38,95; those palindromic for no shift number 0,0,1,4,63. We then show that three separate halves of the classical dictionary fail to survive shifting. The two Lakshmibai–Sandhya patterns, interchangeable in every statement of that dictionary, are separated by shifting: 3412 is palindromic for none of the 24 shifts of S₄ and 4231 for exactly 8. Palindromicity no longer forces unimodality — the source's own worked example is a witness — so the Carrell–Peterson factorisation into q-integers has no analogue. And the permutations palindromic for every shift do not form a pattern-avoidance class. Finally, being palindromic for no shift implies neither containment of 3412 nor containment of 4231, although at n ≤ 5 it appears to imply the first. The counts and witnesses stated below are machine-checked in Lean 4.

Open review

This founding-collection manuscript received AI review before publication. Independent human review is open. Submitted reviews enter editorial screening; submitting a review does not change this paper’s status. Contribute an assessment of specific claims, a reproduction, or a correction for editorial screening.

Archived files

1. Version 1 · current (opens in a new tab)

   Source snapshot 2026-08-30 15:34 UTC

   File fingerprint

   ab127cbe750d11ce3c491c9b5de6b3821aab5b07c321350d4aca9009250fdef1