Partition stabilization and semigroup gaps for ternary Parikh matrices

Christopher Yamba

Research preprint · Version 1

Partition stabilization and semigroup gaps for ternary Parikh matrices

Christopher Yamba

First published 19 October 2026, 23:04:21 CEST (21:04:21 UTC)

Exact counting formulas for words with repeated letters, connecting ternary Parikh matrices to integer partitions and numerical-semigroup gaps.

Abstract

We study words with fixed positive content $(a,b,c)$ over $1<2<3$ , jointly by their Parikh-matrix entries and ordinary inversions. Words with at most $b$ inversions correspond bijectively to pairs of restricted partitions. This gives a bivariate stable product, exact matrix-class sizes in a corner chamber, and explicit first corrections in both inversion number and the deficit $abc-|w|_{123}$ . Below the sharp threshold $(b-1)\min(a,c)+a+c-1$ , an integer deficit is attainable exactly when it belongs to the semigroup generated by $a$ and $c$ . For coprime endpoints and an explicit sufficient bound on $b$ , all missing triple counts are the reflections of the Sylvester gaps. We also evaluate the first exterior matrix fibres and classify the first words containing a reverse triple. The proofs use endpoint gaps, identical-letter shuffles and constructive profile realizations. We distinguish the results proved in Lean from the larger collection established by human-readable proofs.

What the paper determines

A bijection between words with few inversions and pairs of restricted partitions, with exact joint counting formulas.
Exact sizes of specified ternary Parikh-matrix classes, including boundary corrections and the first exterior cases.
A numerical-semigroup description of attainable triple deficits, with conditions determining every missing triple count.

The complete statements, hypotheses, proofs and three vector illustrations are in the 15-page paper.

Proofs and verification

All stated results have written proofs. The downloadable verification package contains computational checks and a partial Lean 4 certificate; the paper specifies which results are formalized. It also records its use of AI assistance.

This is an author-published preprint and has not been peer reviewed. It builds on the author's 2006 thesis and the literature cited in the manuscript.

Suggested citation

Christopher Yamba. Partition stabilization and semigroup gaps for ternary Parikh matrices. Research preprint, version 1, 19 October 2026. https://www.christopheryamba.com/math/ternary-parikh/