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Denote by $SSYT(\lambda, [n])$ the set of all semi-standard Young tableaux of shape lambda with entries in $[n]=\{1, \ldots, n\}$. Denote by $SSYT(\lambda, \infty)$ the set of all semi-standard Young tableaux of shape $\lambda$ with entries in $\{1, 2, \ldots, \infty \}$.

Bender-Knuth involution is described in the lecture notes in Section 2.2. It seems that the definition gives an involution on $SSYT(\lambda, \infty)$. Is there some reference about Bender-Knuth involution on $SSYT(\lambda, [n])$, $n\in \mathbb{Z}_{>0}$? Thank you very much.

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    $\begingroup$ Have you read the original paper? I think it makes it pretty clear why the involution doesn't "overflow" the entries ("column $j$ has a $k+1$ but no $k$" shows that $k+1$ is not out of bound). $\endgroup$
    – darij grinberg
    Commented Dec 14, 2019 at 15:43
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    $\begingroup$ As Darij says, the exact same definitions work for bounded entries; it's just that if the entries are bounded by $n$, then we only have $n-1$ involutions $BK_i$ for $1\leq i \leq n-1$. $\endgroup$
    – Sam Hopkins
    Commented Dec 14, 2019 at 15:44

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