Foulkes characters, Eulerian idempotents, and an amazing matrix
نویسندگان
چکیده
منابع مشابه
Foulkes Characters, Eulerian Idempotents, and an Amazing Matrix
John Holte [17] introduced a family of “amazing matrices” which give the transition probabilities of “carries” when adding a list of numbers. It was subsequently shown that these same matrices arise in the combinatorics of the Veronese embedding of commutative algebra [4, 7, 8] and in the analysis of riffle shuffling [7, 8]. We find that the left eigenvectors of these matrices form the Foulkes ...
متن کاملCarries, Shuffling, and an Amazing Matrix
For this example, 19/40=47.5% of the columns have a carry of 1. Holte shows that if the binary digits are chosen at random, uniformly, in the limit 50% of all the carries are zero. This holds no matter what the base. More generally, if one adds n integers (base b) that are produced by choosing their digits uniformly at random in {0, 1, . . . , b− 1}, the sequence of carries κ0 = 0, κ1, κ2, . . ...
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We prove combinatorial rules that give the minimal and maximal partitions labelling the irreducible constituents of a family of characters for the symmetric group that generalize Foulkes permutation characters. Restated in the language of symmetric functions, our results determine all minimal and maximal partitions that label Schur functions appearing in the plethysms sν ◦ s(m). As a corollary ...
متن کاملCombinatorics of Balanced Carries
We study the combinatorics of addition using balanced digits, deriving an analog of Holte’s “amazing matrix” for carries in usual addition. The eigenvalues of this matrix for base b balanced addition of n numbers are found to be 1, 1/b, · · · , 1/b, and formulas are given for its left and right eigenvectors. It is shown that the left eigenvectors can be identified with hyperoctahedral Foulkes c...
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Phil Hanlon proved that the coefficients of the chromatic polynomial of a graph G are equal (up to sign) to the dimensions of the summands in a Hodge-type decomposition of the top homology of the coloring complex for G. We prove a type B analogue of this result for chromatic polynomials of signed graphs using hyperoctahedral Eulerian idempotents.
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ژورنال
عنوان ژورنال: Journal of Algebraic Combinatorics
سال: 2012
ISSN: 0925-9899,1572-9192
DOI: 10.1007/s10801-012-0343-7