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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 2,1
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COMMENTS
| a(n) is also equal to the sum over all permutations p in S(n) of the number of elements in the set {(i, j): 0 < i < j < n+1 and |i - j| = |p(i) - p(j)|}.
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REFERENCES
| 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
| 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
| 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
| seq(n!*(2*n-1)/3, n=2..20);
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CROSSREFS
| Cf. A006157.
Sequence in context: A108490 A165817 A000172 * A191277 A093303 A199163
Adjacent sequences: A097968 A097969 A097970 * A097972 A097973 A097974
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KEYWORD
| nonn
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AUTHOR
| Emeric Deutsch and Ira Gessel (deutsch(AT)duke.poly.edu), Sep 07 2004
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