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Euler bidiagonal rows with varying parameters; Lah numbers of higher level 2 - #2543
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…level 2 Rows of X + (θ + a_m)(θ + b_m) with positive parameters and nondecreasing a_m + b_m have degree m and negative simple roots, and consecutive rows strictly interlace without common roots (isNegativeSimple_rowsVar, natDegree_rowsVar, strictInterl_rowsVar_succ). The proof adds one operator identity (step_sub_step) to the existing fixed-parameter step theorem; strictInterl_rows becomes a corollary with its statement unchanged. The operator theory moves to EulerBidiagonal/Basic.lean. Application: the Lah numbers of higher level 2 (HigherLah, arXiv:2510.26842). Closes #2541. Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>
…idiagonal.Basic Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>
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Closes #2541.
Theorem (
RealRooted.EulerBidiagonal): fora b : ℕ → ℝwith0 < a m,0 < b manda m + b m ≤ a (m+1) + b (m+1), the rowsrowsVar a b(F 0 = 1,F (m+1) = step (a m) (b m) (F m), i.e.X + (θ + a m)(θ + b m)) satisfyisNegativeSimple_rowsVar,natDegree_rowsVar(degreem),strictInterl_rowsVar_succ:F mstrictly interlacesF (m+1), with no common root.No condition on the products
a m · b mis needed. The proof adds one operator identity,step_sub_step : step c d p − step a b p = C (c + d − (a + b)) * θ p + C (c d − a b) * p, to the existing fixed-parameter step theorem.strictInterl_rowsbecomes a two-line corollary with its statement unchanged; the operator theory moves toEulerBidiagonal/Basic.lean.Application (
RealRooted.HigherLah): the Lah numbers of higher level 2,T(n,k) = T(n−1,k−1) + (n−1+k)² T(n−1,k)(ordered pairs of partitions of [n] into k lists with the same block minima; arXiv:2510.26842). The rows divided byXsatisfy the theorem witha m = b m = m + 2. A second, unrelated instance (alternating parameters with constant sum) is a regression example in the tactic examples.This covers every triangle
A(m+1,k) = A(m,k−1) + (k + a_m)(k + b_m) A(m,k)with a nondecreasing positive suma_m + b_m; compare B.-X. Zhu, Proc. AMS 142 (2014), Thm 2.4, which has fixed parameters and no constant term.Checks:
lake build RealRooted.Production RealRooted.Tactic.Examplespassed; source guards pass; all 19 pre-existing public declaration headers unchanged.🤖 Generated with Claude Code