4 research outputs found

    Poly-Bernoulli numbers and lonesum matrices

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    A lonesum matrix is a matrix that can be uniquely reconstructed from its row and column sums. Kaneko defined the poly-Bernoulli numbers Bm(n)B_m^{(n)} by a generating function, and Brewbaker computed the number of binary lonesum m×nm\times n-matrices and showed that this number coincides with the poly-Bernoulli number Bm(n)B_m^{(-n)}. We compute the number of qq-ary lonesum m×nm\times n-matrices, and then provide generalized Kaneko's formulas by using the generating function for the number of qq-ary lonesum m×nm\times n-matrices. In addition, we define two types of qq-ary lonesum matrices that are composed of strong and weak lonesum matrices, and suggest further researches on lonesum matrices. \Comment: 27 page

    On q-poly-Bernoulli numbers arising from combinatorial interpretations

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    Diagrams of affine permutations and their labellings

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    Thesis (Ph. D.)--Massachusetts Institute of Technology, Department of Mathematics, 2013.Cataloged from PDF version of thesis.Includes bibliographical references (pages 63-64).We study affine permutation diagrams and their labellings with positive integers. Balanced labellings of a Rothe diagram of a finite permutation were defined by Fomin- Greene-Reiner-Shimozono, and we extend this notion to affine permutations. The balanced labellings give a natural encoding of the reduced decompositions of affine permutations. We show that the sum of weight monomials of the column-strict balanced labellings is the affine Stanley symmetric function which plays an important role in the geometry of the affine Grassmannian. Furthermore, we define set-valued balanced labellings in which the labels are sets of positive integers, and we investigate the relations between set-valued balanced labellings and nilHecke words in the nilHecke algebra. A signed generating function of column-strict set-valued balanced labellings is shown to coincide with the affine stable Grothendieck polynomial which is related to the K-theory of the affine Grassmannian. Moreover, for finite permutations, we show that the usual Grothendieck polynomial of Lascoux-Schiitzenberger can be obtained by flagged column-strict set-valued balanced labellings. Using the theory of balanced labellings, we give a necessary and sufficient condition for a diagram to be a permutation diagram. An affine diagram is an affine permutation diagram if and only if it is North-West and admits a special content map. We also characterize and enumerate the patterns of permutation diagrams.by Taedong Yun.Ph.D
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