Math
Short, compact entries: one per piece of work, with a quick summary and the links. The longer thinking usually lives in the blog.
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The beauty of cancellation: height-two positivity, checked in Lean
An illustrated recreational-mathematics journey from a three-bead coloring puzzle to positive tableau survivors. Established theorem and insertion by Cho–Huh; completed all-finite-alphabet Lean formalization in Yamba AI Math.
Read the illustrated article · Full technical article (PDF, 11 pages) · Lean theorem ↗
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Four diagonals suffice: completeness of Parikh matrices when letters occur at most twice
If every letter other than the first and the last occurs at most twice, the first four diagonals determine the Parikh matrix. A sharpness family with ballot and Catalan numbers, a conjecture for higher multiplicities, and Lean 4 proofs of the main theorem.
Article and abstract · PDF · Source, Lean and checks · Repository ↗
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From Euler numbers to the golden ratio: rigidity, ranges and the most ambiguous Parikh matrices for words with repeated letters
Which Parikh matrices determine their words, which values their entries can take, and which are shared by the most words, for words with repeated letters. Research preprint, 55 pages, with written proofs, eight figures and Python checks.
Article and abstract · PDF · Source and checks · Repository ↗
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Partition stabilization and semigroup gaps for ternary Parikh matrices
Exact counting formulas for words with repeated letters, connecting ternary Parikh matrices to integer partitions and numerical-semigroup gaps. Research preprint with complete written proofs, three figures and a partial Lean certificate.
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Purple Square Atlas
Six purple-and-white tiles on a 2 × 2 board give 1,296 oriented arrangements. The four rotations fix 1,296, 6, 36 and 6 of them, so Burnside's lemma gives (1,296 + 6 + 36 + 6) / 4 = 336 rotation classes. The catch: turning the board also turns the diagonal inside every tile.
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The Ouroboros Problem
A check that cannot fail carries no evidence. The same loop runs from logic, where a circular proof is a null result, through statistics (data leakage, double dipping) to AI agents that do science, with design rules for agents that cannot cheat.
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Counting Melodies
A bar of music, counted exactly: rhythm becomes integer compositions, the no-two-rests rule becomes a regular language, and the language becomes a rational generating function. Revision 2.
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Parikh Matrices and Permutation Statistics
A word's Parikh matrix counts its scattered subwords. Counting the permutations that share one gives exactly the descent statistic, a bridge between formal languages and permutation statistics. Degree thesis, Stockholm University 2006:8.
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Combinatorial Mathematical Music Theory
Musical sets and mappings, multiset permutations, descent statistics, Parikh matrices and q-analogues. The earliest antecedent of Counting Melodies. Unpublished manuscript.