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Search: id:A097971
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| A097971 |
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Number of alternating runs in all permutations of [n] (the permutation 732569148 has four alternating runs: 732, 2569, 91, and 148). |
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2
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| 2, 10, 56, 360, 2640, 21840, 201600, 2056320, 22982400, 279417600, 3672345600, 51891840000, 784604620800, 12640852224000, 216202162176000, 3912561709056000, 74694359900160000, 1500289571708928000
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OFFSET
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2,1
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REFERENCES
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M. Bona, Combinatorics of Permutations, Chapman & Hall/CRC, Boca Raton, FL, 2004, pp. 24-30.
D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, 1973, Vol. 3, pp. 46 and 587-8.
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FORMULA
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a(n)=n!(2n-1)/3. E.g.f. = x^2*(3-x)/[3(1-x)^2]. a(n)=2*A006157
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EXAMPLE
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a(3)=10 because the permutations 123, 132, 312, 213, 231, 321 have the following alternating runs: 123, 13, 32, 31, 12, 21, 13, 23, 31, and 321.
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MAPLE
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seq(n!*(2*n-1)/3, n=2..20);
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CROSSREFS
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Cf. A006157.
Sequence in context: A122826 A108490 A000172 this_sequence A093303 A075870 A074608
Adjacent sequences: A097968 A097969 A097970 this_sequence A097972 A097973 A097974
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KEYWORD
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nonn
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AUTHOR
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Emeric Deutsch and Ira Gessel (deutsch(AT)duke.poly.edu), Sep 07 2004
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