bibkey: mateescu2001parikh authors: Alexandru Mateescu; Arto Salomaa; Kai Salomaa; Sheng Yu year: 2001 title: A sharpening of the Parikh mapping doi: 10.1051/ita:2001131 url: https://www.numdam.org/item/ITA_2001__35_6_551_0/ claim: “Definition 1.1 and Theorem 2.1 define the ordered Parikh matrix product and identify its upper entries with scattered-subword counts.” strata_touched:
- D5/S3/Observer/GoldenChronology/BinaryParikhStepTwoBridge license: citation-only triage: anchor
A sharpening of the Parikh mapping
Mateescu, A. Salomaa, K. Salomaa and Yu, RAIRO - Theoretical Informatics and Applications 35(6), 551-564, define the Parikh matrix mapping for an ordered alphabet. Definition 1.1 assigns to its j-th letter the identity matrix with one additional entry at (j,j+1), and extends this assignment multiplicatively in word order. Theorem 2.1 identifies the entry (i,j+1) with the number of scattered occurrences of the consecutive alphabet segment from i through j. The diagonal entries are one and the lower entries are zero.
For the binary specialization, the first letter is true and the second is false. Converting the paper’s one-based matrix indices to zero-based indices gives the generators I+E01 and I+E12. The entries (0,1), (1,2) and (0,2) are respectively the true count, false count and scattered true-before-false count. The empty word maps to the identity. Casting these natural counts into the integers does not change the attributed identity.
This note attests the standard product and its entry theorem. The connections to the repository’s factorial Chen signature, doubled Magnus center and count-and-center recovery interface are separate repository derivations. Recursive counting and matrix normalization helpers carry no novelty claim.
Verified locator
- DOI: https://doi.org/10.1051/ita:2001131
- Archive record: https://www.numdam.org/item/ITA_2001__35_6_551_0/
- Original article: https://www.numdam.org/item/ITA_2001__35_6_551_0.pdf
- Definition 1.1 in Section 1; Theorem 2.1 in Section 2, printed page 554.