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What is the origin of the abacus bijection (aka the rim hook bijection, aka the Stanton-White bijection, aka James's bijection)?

Igor Pak, in his 2000 article "Ribbon tile invariants" (Transactions of the American Mathematical Society, volume 352 (2000), pages 5525-5561), summarizes the situation thus: "The theorem goes back to Nakayama and Robinson (see [R], [JK]). In modern times it was rediscovered by Stanton and White (see [SW], [FS]) and is sometimes attributed to them." But I have found no other attribution to Nakayama and Robinson (perhaps because I have no access to [R]). I have however looked at [JK] which contains a statement of the result, attributed to G. James (the book's first author).

Pak is a scrupulous mathematical historian (see for instance his excellent essay https://www.math.ucla.edu/~pak/papers/cathist4.pdf, included as an appendix in Richard Stanley's book on the Catalan numbers), so I'm inclined to take him at his word regarding Nakayama and Robinson. Still, I'd feel better knowing more details.

[FS] S. Fomin, D. Stanton, Rim hook lattices, St. Petersburg Math. J. 9 (1998), 1007–1016.

[JK] G. James, A. Kerber, The Representation Theory of the Symmetric Group, Addison-Wesley, Reading, MA, 1981.

[R] G. de B. Robinson, Representation Theory of the Symmetric Group, Edinburgh University Press and Univ. of Toronto Press, 1961.

[SW] D. Stanton, D. White, A Schensted algorithm for rim hook tableaux, J. Comb. Theory, Ser. A 40 (1985), 211–247.

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  • $\begingroup$ I am not a scrupulous mathematical historian but my belief is that the $p$-core being well defined, hook lengths, etc., were known to Nakayama/Robinson (hook lengths even to Frobenius?), but the nice pictorial representation by abacuses was James. I defer to anyone more knowledgeable about this though. $\endgroup$
    – David A. Craven
    Commented Sep 25, 2022 at 17:35

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After looking at Robinson's book, my coauthors and I settled upon the following wording:

The version of the bijection that we use is due to Gordon James (see \cite{JamesKerber}) but different forms of it seem to have been discovered independently by various people working in the field of modular representation theory around 1950; this community includes H.\ Farahat, J.\ S.\ Frame, D.\ E.\ Littlewood, T.\ Nakayama, M.\ Osima, G.\ de B.\ Robinson, R.\ A.\ Staal, and R.\ M.\ Thrall. The interested reader may find more details in the book \cite{Robinson} and the references it contains.

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