Carries, Shuffling, and an Amazing Matrix
نویسندگان
چکیده
منابع مشابه
Carries, 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, . . ...
متن کاملCarries, shuffling, and symmetric functions
Article history: Received 1 February 2009 Accepted 11 February 2009 Available online 15 April 2009 MSC: 60C05 60J10 05E05
متن کامل9 Carries , Shuffling , and Symmetric Functions
The “carries” when n random numbers are added base b form a Markov chain with an “amazing” transition matrix determined by Holte [24]. This same Markov chain occurs in following the number of descents or rising sequences when n cards are repeatedly riffle shuffled. We give generating and symmetric function proofs and determine the rate of convergence of this Markov chain to stationarity. Simila...
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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 ...
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For any matrix X let X′ denote its transpose. It is known that if A is an n-by-n matrix over a field F , then A and A′ are congruent over F , i.e., XAX′ = A′ for some X ∈ GLn(F ). Moreover, X can be chosen so that X2 = In, where In is the identity matrix. An algorithm is constructed to compute such an X for a given matrix A. Consequently, a new and completely elementary proof of that result is ...
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ژورنال
عنوان ژورنال: American Mathematical Monthly
سال: 2009
ISSN: 0002-9890,1930-0972
DOI: 10.4169/000298909x474864