This paper makes a combinatorial study of the two monoids and the two typesof tableaux that arise from the two possible generalizations of the PatienceSorting algorithm from permutations (or standard words) to words. For bothtypes of tableaux, we present Robinson--Schensted--Knuth-type correspondences(that is, bijective correspondences between word arrays and certain pairs ofsemistandard tableaux of the same shape), generalizing two knowncorrespondences: a bijective correspondence between standard words and certainpairs of standard tableaux, and an injective correspondence between words andpairs of tableaux. We also exhibit formulas to count both the number of each type of tableauxwith given evaluations (that is, containing a given number of each symbol).Observing that for any natural number $n$, the $n$-th Bell number is given bythe number of standard tableaux containing $n$ symbols, we restrict theprevious formulas to standard words and extract a formula for the Bell numbers.Finally, we present a `hook length formula' that gives the number of standardtableaux of a given shape and deduce some consequences.
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机译：本文使两种单套管的组合研究和两种类型的表格，从排列（或标准单词）的图谱算法的两种可能的概括出来。对于Tableaux两种类型，我们呈现罗宾逊 - 桑切尔 - Knuth型对应关系（即字阵之间的怪异对应关系，同一形状的特定成对的特定成对的曲线），概括了两个已知的重复记录：标准单词和某方面之间的基础对应关系标准的TableAux，以及Tableaux的单词之间的注射对应。我们还展示了算法计算每种类型的表款的数量（即包含给定数量的每个符号）。为任何自然数字$ N $，$ N $ -th贝尔号码由包含$ n $符号的标准表格数，我们将不可原形的公式限制为标准单词并提取铃声的公式。最后，我们呈现了一个“钩子长度公式”，它给出了给定形状的标准化术数量并推断出一些后果。
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