| Résume | Once upon a time, there was a mathematical journal which was "almost'' a journal in the
sense that one could find there either papers in final form, or reports of talks given at a local
(but well-known and welcoming international researchers) Seminar, or papers that were not
necessarily in final form and could even stay forever in an almost unfinished state. This journal
was the Séminaire de Théorie des Nombres de Bordeaux, that became later a more
classical journal, the Journal de Théorie des Nombres de Bordeaux.
Among the many papers that were not in a final form in the Séminaire (and that were
not necessarily followed by a "real paper'' appeared later elsewhere), one can find a quite
modest paper entitled "Suites infinies à répétitions bornées'' (in the 1983--1984 issue).
In that paper there was a short study of the number of cycles of a family of permutations
related to the Toeplitz transform of sequences.
Up to notation the number of cycles of the same family of permutations appeared in 2021 in
a paper by Y. Guo, G. Han, and W. Wu about "apwenian sequences''.
The formulas given in these two papers were distinct, though---of course---equivalent.
Thus it was tempting to prove directly their equality.
This possibly frivolous question lead to unexpected discoveries: the family of permutations
above or a variation of it are linked to numerous subjects, not only in mathematics (combinatorics
of words, dynamical systems, number theory, correcting algorithms), but also in card-shuffling,
juggling, change-ringing, poetry, and music composition.
The details can be found in a paper by J.-P. A., M. Stipulanti, and J.-Y. Yao, appeared in
The Mathematical Intelligencer in 2026. |
