Tableau stabilization and rectangular tableaux fixed by promotion powers (Q2038903)

In this paper, the author introduces tableau stabilization, a new phenomenon and statistic on Young tableaux based on jeu de taquin. For any standard skew tableau \(S\), let \(S^{(k)}\) be the result of attaching \((k-1)\) shifted copies of \(S\) to the right of \(S\) so that the result is a standard skew tableau. If \(m\) is the size of \(S\) then \(\operatorname{stab}(S)\) is the minimum positive integer \(k\) such that the entries in \([(k - 1)m + 1, km]\) lie in the same rows in \(\mathrm{Rect}(S^{(k)})\) and \(S^{(k)}\), where \(\mathrm{Rect}(S^{(k)})\) is the rectification operator. For example, if \[ S =\quad \begin{tabular}{| c | c | c | c |} \cline {3-4} \multicolumn{2}{ c }{} \vline & 1 & 3 \\ \cline {3-4} \multicolumn{2}{ c }{} \vline & 5 & 6 \\ \cline{3-4} \cline{1-2} 2 & 4 & \multicolumn{2}{ c }{} \\ \cline{1-2} \end{tabular} \] then \[ S^{(3)} = \quad \begin{tabular}{| c | c | c | c | c | c | c | c |} \cline {3-8} \multicolumn{2}{ c }{} \vline & 1 & 3 & 7 & 9 & 13 & 15 \\ \cline {3-8} \multicolumn{2}{ c }{} \vline & 5 & 6 & 11 & 12 & 17 & 18 \\ \cline{3-8} \cline{1-6} 2 & 4 & 8 & 10 & 14 & 16 & \multicolumn{2}{ c }{} \\ \cline{1-6} \end{tabular} \] and \[ \mathrm{Rect}(S^{(3)}) =\quad \begin{tabular}{| c | c | c | c | c | c | c | c |} \hline 1 & 3 & 5 & 6 & 7 & 9 & 13 & 15 \\ \hline 2 & 4 & 11 & 12 & 17 & 18 & \multicolumn{2}{ c }{} \\ \cline{1-6} 8 & 10 & 14 & 16 & \multicolumn{4}{ c }{} \\ \cline{1-4} \end{tabular} \] \(\operatorname{stab}(S) = 2\) as \(2\) does not lie in the same row in \(S^{(3)}\) and \(\mathrm{Rect}(S^{(3)})\), but \(7, 8,9,10,11,12\) do. In Theorem 3.6, the author shows that stabilization is constant on dual equivalence classes. He also proves, in Lemma 3.9, that once a skew tableau stabilizes, it continues to stabilize, and any skew tableau must stabilize eventually. He conjectures that any standard skew tableau with \(b\) rows and decreasing row sizes stabilizes at \(b\) but was unable to prove it unless for standard skew tableau with \(b\) rows of the same size. However, he shows in Theorem 1.5 that any standard skew tableau with \(b \geq 2\) rows and decreasing row sizes stabilizes at \(2b - 2\). In Sections 6 and 7, tableau stabilization is employed to explicitly construct and describe promotion on \(\mathrm{SYT}((ar)^b)^{p^{br}}\) for \(a \geq 2b - 1\), and \(\mathrm{SYT}((2r)^b)^{p^{br}}\). In Section 8, stabilization is studied as a permutation statistic and in Section 9, open problems are presented.