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Cigler’s Narayana Strip Product Formula

Abstract

The product of two signed Narayana strip series is the Narayana strip series with squared parameters at heights 4m and 4m + 1.

Theorem 1.1 (The product formula at both strip heights).

Lean statement: D5/S3/Combinatorics/NarayanaStrip/CiglerStripProduct.result

Proof. Machine-checked in Lean as D5/S3/Combinatorics/NarayanaStrip/CiglerStripProduct.result (✓ std3). ∎

Resolves. Problems/cigler-narayana-strip-product (proved) by D5/S3/Combinatorics/NarayanaStrip/CiglerStripProduct.result.

Source. Repository-derived.

Acknowledgement. Johann Cigler (2026). Some sequences and number triangles which are related to Narayana polynomials and to q-Narayana polynomials for q=-1. DOI: 10.48550/arXiv.2608.03363. URL: https://arxiv.org/abs/2608.03363v2.

Commentary.

For every positive integer m and each height h equal to 4m or 4m + 1, C^{(h)}(t^2, z^2) = c^{(h)}(t, z)c^{(h)}(-t, -z) as formal power series in z with coefficients in the integer polynomial ring in t. Here C^{(h)} and c^{(h)} sum Dyck paths confined to heights zero through h, with z marking semilength, every up-step weighted one, and down-step arrival weights repeating 1, t for C^{(h)} and 1, t, -1, -t for c^{(h)}. The equality holds for every coefficient of z and proves Conjecture 2, equation (79), of Cigler’s paper. First-return decomposition gives the continuant representation of each bounded series. The numerator and denominator factorizations at indices h + 1, together with invertibility of the denominators, give the product identity.

References