Wolfram Function Repository
Instant-use add-on functions for the Wolfram Language
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Get the number of permutations with a given number of ascents
ResourceFunction["EulerianNumber"][n,k] gives the number of permutations of {1,2,…,n} with k ascents. |
Count the number of ascents of the 24 permutations of {1,2,3,4}:
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Tally up permutations by the number of ascents:
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EulerianNumber gives the same list, calculated without implicitly listing the individual ascents:
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The numbers form an infinite lower-triangular matrix:
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The row sums are the factorials because they count the number of permutations:
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Here the entries are signed in a checkerboard pattern:
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Here are those row sums:
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Expand Tan[x]:
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Get rid of the factorials in the denominators to match the row sums up to signs and a shift:
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