Narayana Polynomials¶

Package the Narayana distribution of a Catalan family into a polynomial whose coefficient of each power counts objects with a specified statistic, retaining the Catalan total at one while exposing symmetry, unimodality, real-rootedness, and specializations.

Version: v2 · 2026-09-06 · History
Domain-specific #: 2346
Origin domain: mathematics
Subdomain: enumerative combinatorics
Aliases: Narayana polynomial, Type-A Narayana polynomial

Core Idea¶

The Narayana polynomials are finite generating polynomials whose coefficients are the type-A Narayana numbers. For a positive integer n and 1 <= k <= n, the Narayana number is N(n,k) = (1/n) binom(n,k) binom(n,k-1). One common convention is N_n(t) = sum_{k=1}^n N(n,k)t^k, with N_0(t)=1; another shifts the exponent and uses t^(k-1). The two conventions carry the same coefficient triangle but differ by a factor of t, so the exponent convention must be declared before identities, reciprocal polynomials, or constant terms are compared.

Scope of Application¶

Narayana polynomials are literal when a type-A Narayana row is encoded as a statistic-generating polynomial with an explicit size, statistic, and exponent convention.

Dyck paths. Counting semilength-n paths by number of peaks.
Noncrossing partitions. Counting partitions by number of blocks.
Pattern avoidance. Recording descent or related distributions on Catalan permutation classes.
Plane trees. Transporting the coefficient distribution through a stated Catalan bijection.
Polynomial analysis. Studying zeros, interlacing, log-concavity, and gamma expansions.
Generating functions. Combining the size and statistic variables in algebraic series.
Specialization identities. Recovering Catalan, Schröder, or weighted totals under verified conventions.
Bijection design. Proving equidistribution by mapping one statistic-refined Catalan family to another.

Clarity¶

State whether n begins at zero, the allowed range of k, whether N(0,0) is separately defined, and whether the exponent is k or k-1. Give at least one coefficient formula and one combinatorial interpretation. A table should be checked from the definition: for the unshifted convention, N_1(t)=t, N_2(t)=t+t^2, N_3(t)=t+3t^2+t^3, and N_4(t)=t+6t^2+6t^3+t^4.

Manages Complexity¶

A Catalan number compresses a large family to one total. The Narayana polynomial restores one dimension of structure without listing every object: each coefficient is a fiber size of the statistic map, evaluation weights those fibers, differentiation yields statistic moments after normalization, and roots constrain coefficient shape. The representation enables algebraic proofs of distributional facts and bijective transfers between Catalan families. Complexity returns through convention drift and proliferating generalizations.

Abstract Reasoning¶

Choose a Catalan family of size n and a statistic with Narayana distribution. 2. Fix the coefficient convention for N(n,k) and the valid range of k. 3. Choose the exponent convention and record any shift or reciprocal normalization. 4. Map each object with statistic k to the monomial carrying the corresponding power. 5. Sum monomials and group equal powers to obtain the coefficient polynomial.

Knowledge Transfer¶

The abstraction transfers a general enumerative pattern: replace a total count by a polynomial whose exponents mark a statistic and whose coefficients count its fibers. This move turns combinatorial refinement into algebra. It supports moments, symmetry tests, specializations, real-rootedness arguments, and bijections. The Narayana family provides a disciplined exemplar because several Catalan models yield the same row. Transfer requires proving equidistribution; sharing a Catalan total does not guarantee sharing the Narayana polynomial.

Relationships to Other Abstractions¶

Current abstraction Narayana Polynomials Domain-specific

Parents (1) — more general patterns this builds on

Narayana Polynomials is a kind of Representation Prime

Representation is the strict parent by specialization.

Hierarchy path (1) — routes to 1 parentless root

Narayana Polynomials → Representation → Abstraction

Neighborhood in Abstraction Space¶

Narayana Polynomials sits in a sparse region of the domain-specific corpus (97^th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.

Family — Unclustered & Miscellaneous (1565 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-09-08