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Search: id:A005287
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| A005287 |
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Number of permutations of [n] with four inversions.
(Formerly M3894)
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+0
3
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| 5, 20, 49, 98, 174, 285, 440, 649, 923, 1274, 1715, 2260, 2924, 3723, 4674, 5795, 7105, 8624, 10373, 12374, 14650, 17225, 20124, 23373, 26999, 31030, 35495, 40424, 45848, 51799, 58310, 65415, 73149, 81548, 90649, 100490, 111110, 122549, 134848
(list; graph; listen)
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OFFSET
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4,1
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 255, #2, b(n,4).
F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 241.
R. K. Guy, personal communication.
R. H. Moritz and R. C. Williams, A coin-tossing problem and some related combinatorics, Math. Mag., 61 (1988), 24-29.
E. Netto, Lehrbuch der Combinatorik. 2nd ed., Teubner, Leipzig, 1927, p. 96.
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 1, 1999; see Exercise 1.30, p. 49.
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LINKS
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S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
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FORMULA
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n*(n+1)*(n^2+n-14)/24
binomial(n,4)+binomial(n,3)-binomial(n,2), n>=5. - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 23 2006
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EXAMPLE
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[2, 4, 3, 1], [3, 2, 4, 1], [3, 4, 1, 2], [4, 1, 3, 2], [4, 2, 1, 3] have 4 inversions.
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MAPLE
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[seq(binomial(n, 4)+binomial(n, 3)-binomial(n, 2), n=5..43)]; - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 23 2006
A005287:=(-5+5*z+z**2-3*z**3+z**4)/(z-1)**5; [Conjectured by S. Plouffe in his 1992 dissertation.]
seq(sum(binomial(n, m), m=1..4)-n^2, n=5..43); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 19 2008
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PROGRAM
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(PARI) a(n)=if(n<4, 0, n*(n+1)*(n^2+n-14)/24)
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CROSSREFS
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Cf. A008302.
Sequence in context: A168011 A160749 A147002 this_sequence A147488 A134481 A062158
Adjacent sequences: A005284 A005285 A005286 this_sequence A005288 A005289 A005290
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KEYWORD
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nonn,easy
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com), Robert G. Wilson v (rgwv(AT)rgwv.com)
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EXTENSIONS
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Additional comments from Michael Somos, Jun 25, 2002.
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