4 research outputs found
Poly-Bernoulli numbers and lonesum matrices
A lonesum matrix is a matrix that can be uniquely reconstructed from its row
and column sums. Kaneko defined the poly-Bernoulli numbers by a
generating function, and Brewbaker computed the number of binary lonesum
-matrices and showed that this number coincides with the
poly-Bernoulli number . We compute the number of -ary lonesum
-matrices, and then provide generalized Kaneko's formulas by using
the generating function for the number of -ary lonesum -matrices.
In addition, we define two types of -ary lonesum matrices that are composed
of strong and weak lonesum matrices, and suggest further researches on lonesum
matrices. \Comment: 27 page
Diagrams of affine permutations and their labellings
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