Coefficients of the Narayana Strip Product

Abstract

The product formula for bounded Narayana series is expressed as equality of polynomial coefficients at heights 4m and 4m + 1.

Definition 1.1 (The squared-parameter coefficient).

Lean statement: D5/S3/Combinatorics/NarayanaStrip/CiglerStripProductDefs.lhsCoeff

Formalization. D5/S3/Combinatorics/NarayanaStrip/CiglerStripProductDefs.lhsCoeff (✓ std3).

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 nonnegative integers h and N, the coefficient of z^N in C^{(h)}(t^2, z^2) is zero when N is odd. When N is even, it is the Narayana-weighted sum over Dyck paths of semilength N/2 confined to heights zero through h, with t replaced by t^2. Each up-step has weight one, and each down-step arriving at height k has weight one for even k and t for odd k before this substitution.

Definition 1.2 (The signed convolution coefficient).

Lean statement: D5/S3/Combinatorics/NarayanaStrip/CiglerStripProductDefs.rhsCoeff

Formalization. D5/S3/Combinatorics/NarayanaStrip/CiglerStripProductDefs.rhsCoeff (✓ std3).

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 nonnegative integers h and N, the coefficient of z^N in c^{(h)}(t, z)c^{(h)}(-t, -z) is the sum over i from zero through N of c_i^{(h)}(t) times (-1)^(N - i) times c_{N-i}^{(h)}(-t). Here c_j^{(h)}(t) is the weighted sum over Dyck paths of semilength j confined to heights zero through h, with up-step weight one and down-step arrival weights repeating 1, t, -1, -t.

Definition 1.3 (The two strip product identities).

Lean statement: D5/S3/Combinatorics/NarayanaStrip/CiglerStripProductDefs.claim

Formalization. D5/S3/Combinatorics/NarayanaStrip/CiglerStripProductDefs.claim (✓ std3).

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 every nonnegative integer N, the coefficient of z^N in C^{(4m)}(t^2, z^2) equals the coefficient of z^N in c^{(4m)}(t, z)c^{(4m)}(-t, -z), and the same coefficient equality holds at height 4m + 1. Both equalities are identities of polynomials in t with integer coefficients. This is the coefficientwise form of Conjecture 2, equation (79), of Cigler’s paper.

References

• Truth anchor: D5/S3/Combinatorics/NarayanaStrip/CiglerStripProductDefs.claim
• Truth anchor: D5/S3/Combinatorics/NarayanaStrip/CiglerStripProductDefs.lhsCoeff
• Truth anchor: D5/S3/Combinatorics/NarayanaStrip/CiglerStripProductDefs.rhsCoeff
• Dependency: D5/S3/Combinatorics/NarayanaStrip/CiglerStripExpansionDefs