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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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A bijective proof of Stanley’s shuffling theorem
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by I. P. Goulden PDF
Trans. Amer. Math. Soc. 288 (1985), 147-160 Request permission


For two permutations $\sigma$ and $\omega$ on disjoint sets of integers, consider forming a permutation on the combined sets by "shuffling" $\sigma$ and $\omega$ (i.e., $\sigma$ and $\omega$ appear as subsequences). Stanley [10], by considering $P$-partitions and a $q$-analogue of Saalschutz’s $_3{F_2}$ summation, obtained the generating function for shuffles of $\sigma$ and $\omega$ with a given number of falls (an element larger than its successor) with respect to greater index (sum of positions of falls). It is a product of two $q$-binomial coefficients and depends only on remarkably simple parameters, namely the lengths, numbers of falls and greater indexes of $\sigma$ and $\omega$. A combinatorial proof of this result is obtained by finding bijections for lattice path representations of shuffles which reduce $\sigma$ and $\omega$ to canonical permutations, for which a direct evaluation of the generating function is given.
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Additional Information
  • © Copyright 1985 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 288 (1985), 147-160
  • MSC: Primary 05A15; Secondary 05A30, 06A99
  • DOI:
  • MathSciNet review: 773053