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One Classifier for All Ambiguous Words: Overcoming Data Sparsity by Utilizing Sense Correlations Across Words
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Abstract Kreweras words are words consisting of n $$\mathrm {A}$$ A ’s, n $$\mathrm {B}$$ B ’s, and n $$\mathrm {C}$$ C ’s in which every prefix has at least as many $$\mathrm {A}$$ A ’s as $$\mathrm {B}$$ B ’s and at least as many $$\mathrm {A}$$ A ’s as $$\mathrm {C}$$ C ’s. Equivalently, a Kreweras word is a linear extension of the poset $$\mathsf{V}\times [n]$$ V × [ n ] . Kreweras words were introduced in 1965 by Kreweras, who gave a remarkable product formula for their enumeration. Subsequently they became a fundamental example in the theory of lattice walks in the quarter plane. We study Schützenberger’s promotion operator on the set of Kreweras words. In particular, we show that 3 n applications of promotion on a Kreweras word merely swaps the $$\mathrm {B}$$ B ’s and $$\mathrm {C}$$ C ’s. Doing so, we provide the first answer to a question of Stanley from 2009, asking for posets with ‘good’ behavior under promotion, other than the four families of shapes classified by Haiman in 1992. We also uncover a strikingly simple description of Kreweras words in terms of Kuperberg’s $$\mathfrak {sl}_3$$ sl 3 -webs, and Postnikov’s trip permutation associated with any plabic graph. In this description, Schützenberger’s promotion corresponds to rotation of the web.more » « less
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