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The largest singletons in weighted set partitions and its applications

Abstract : Recently, Deutsch and Elizalde studied the largest fixed points of permutations. Motivated by their work, we consider the analogous problems in weighted set partitions. Let A (n,k) (t) denote the total weight of partitions on [n + 1] = \1,2,..., n + 1\ with the largest singleton \k + 1\. In this paper, explicit formulas for A (n,k) (t) and many combinatorial identities involving A (n,k) (t) are obtained by umbral operators and combinatorial methods. In particular, the permutation case leads to an identity related to tree enumerations, namely, [GRAPHICS] where D-k is the number of permutations of [k] with no fixed points.
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yidong Sun, yanjie Xu. The largest singletons in weighted set partitions and its applications. Discrete Mathematics and Theoretical Computer Science, DMTCS, 2011, Vol. 13 no. 3 (3), pp.75--86. ⟨10.46298/dmtcs.535⟩. ⟨hal-00990479⟩

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