{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,22]],"date-time":"2026-01-22T19:48:45Z","timestamp":1769111325771,"version":"3.49.0"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>The RSK correspondence generalises the Robinson-Schensted correspondence by replacing permutation matrices by matrices with entries in ${\\bf N}$, and standard Young tableaux by semistandard ones. For $r\\in{\\bf N}_{&gt;0}$, the Robinson-Schensted correspondence can be trivially extended, using the $r$-quotient map, to one between $r$-coloured permutations and pairs of standard $r$-ribbon tableaux built on a fixed $r$-core (the Stanton-White correspondence). Viewing $r$-coloured permutations as matrices with entries in ${\\bf N}^r$ (the non-zero entries being unit vectors), this correspondence can also be generalised to arbitrary matrices with entries in ${\\bf N}^r$ and pairs of semistandard $r$-ribbon tableaux built on a fixed $r$-core; the generalisation is derived from the RSK correspondence, again using the $r$-quotient map. Shimozono and White recently defined a more interesting generalisation of the Robinson-Schensted correspondence to $r$-coloured permutations and standard $r$-ribbon tableaux; unlike the Stanton-White correspondence, it respects the spin statistic on standard $r$-ribbon tableaux, relating it directly to the colours of the $r$-coloured permutation. We define a construction establishing a bijective correspondence between general matrices with entries in ${\\bf N}^r$ and pairs of semistandard $r$-ribbon tableaux built on a fixed $r$-core, which respects the spin statistic on those tableaux in a similar manner, relating it directly to the matrix entries. We also define a similar generalisation of the asymmetric RSK correspondence, in which case the matrix entries are taken from $\\{0,1\\}^r$.   More surprising than the existence of such a correspondence is the fact that these Knuth correspondences are not derived from Schensted correspondences by means of standardisation. That method does not work for general $r$-ribbon tableaux, since for $r\\geq3$, no $r$-ribbon Schensted insertion can preserve standardisations of horizontal strips. Instead, we use the analysis of Knuth correspondences by Fomin to focus on the correspondence at the level of a single matrix entry and one pair of ribbon strips, which we call a shape datum. We define such a shape datum by a non-trivial generalisation of the idea underlying the Shimozono-White correspondence, which takes the form of an algorithm traversing the edge sequences of the shapes involved. As a result of the particular way in which this traversal has to be set up, our construction directly generalises neither the Shimozono-White correspondence nor the RSK correspondence: it specialises to the transpose of the former, and to the variation of the latter called the Burge correspondence.   In terms of generating series, our shape datum proves a commutation relation between operators that add and remove horizontal $r$-ribbon strips; it is equivalent to a commutation relation for certain operators acting on a $q$-deformed Fock space, obtained by Kashiwara, Miwa and Stern. It implies the identity $$\\sum_{\\lambda\\geq_r(0)}G^{(r)}_\\lambda(q^{1\\over2},X) G^{(r)}_\\lambda(q^{1\\over2},Y) =\\prod_{i,j\\in{\\bf N}}\\prod_{k=0}^{r-1}{1\\over1-q^kX_iY_j}; $$ where $G^{(r)}_\\lambda(q^{1\\over2},X)\\in{\\bf Z}[q^{1\\over2}][[X]]$ is the generating series by $q^{{\\rm spin}(P)}X^{{\\rm wt}(P)}$ of semistandard $r$-ribbon tableaux $P$ of shape $\\lambda$; the identity is a $q$-analogue of an $r$-fold Cauchy identity, since the series factors into a product of $r$ Schur functions at $q^{1\\over2}=1$. Our asymmetric correspondence similarly proves $$\\sum_{\\lambda\\geq_r(0)}G^{(r)}_\\lambda(q^{1\\over2},X) \\check G^{(r)}_\\lambda(q^{1\\over2},Y) =\\prod_{i,j\\in{\\bf N}}\\prod_{k=0}^{r-1}(1+q^kX_iY_j). $$ with $\\check G^{(r)}_\\lambda(q^{1\\over2},X)$ the generating series by $q^{{\\rm spin}^{\\rm t}(P)}X^{{\\rm wt}(P)}$ of transpose semistandard $r$-ribbon tableaux $P$, where ${\\rm spin}^{\\rm t}(P)$ denotes the spin as defined using the standardisation appropriate for such tableaux.<\/jats:p>","DOI":"10.37236\/1907","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T05:19:14Z","timestamp":1578719954000},"source":"Crossref","is-referenced-by-count":9,"title":["Spin-Preserving Knuth Correspondences for Ribbon Tableaux"],"prefix":"10.37236","volume":"12","author":[{"given":"Marc A. A.","family":"Van Leeuwen","sequence":"first","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2005,2,14]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v12i1r10\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v12i1r10\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,18]],"date-time":"2020-01-18T04:51:07Z","timestamp":1579323067000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v12i1r10"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2005,2,14]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2005,1,7]]}},"URL":"https:\/\/doi.org\/10.37236\/1907","relation":{},"ISSN":["1077-8926"],"issn-type":[{"value":"1077-8926","type":"electronic"}],"subject":[],"published":{"date-parts":[[2005,2,14]]},"article-number":"R10"}}