Two new papers on Parikh matrices, and a theorem checked in Lean
Two new preprints continue my 2006 thesis on Parikh matrices and permutation statistics.
Four diagonals suffice answers a question the thesis raised by reading the Parikh matrix diagonal by diagonal. In the thesis, letters occur once, and the pair counts on the third diagonal determine the whole matrix. In general no fixed number of diagonals is enough. The new paper proves that the first four diagonals suffice when every letter, except the first and the last, occurs at most twice. A family of examples shows the pattern cannot simply continue: with n copies of each inner letter, counts of length n are not enough. Those examples have ballot numbers as their counts, with the Catalan numbers among them. The main theorem is proved in Lean 4, at the level of words, with no gaps.
From Euler numbers to the golden ratio is a longer, 55-page study. It asks which Parikh matrices determine their words and which values their entries can take. It also asks which matrices are shared by the most words, for words with repeated letters. The answers run from the Euler numbers of alternating permutations to a threshold asymptotic to the golden ratio.
Both article pages include the PDF, the LaTeX source and a verification package. The sources, the Lean proofs and the checks are also in the Yamba AI Math repository. Generative AI tools assisted with the exploration, the proofs, the Lean formalisation and the writing, as each paper records. The mathematics and its responsibility are mine. These are author-published preprints and have not been peer reviewed.