A005283 Number of permutations of (1,...,n) having n-5 inversions (n>=5).
1, 5, 20, 76, 285, 1068, 4015, 15159, 57486, 218895, 836604, 3208036, 12337630, 47572239, 183856635, 712033264, 2762629983, 10736569602, 41788665040, 162869776650, 635562468075, 2482933033659, 9710010151831, 38008957336974, 148912655255315, 583885852950802
Offset: 5
Keywords
Examples
a(6)=5 because we have 213456, 132456, 124356, 123546, and 123465.
References
- F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 241.
- S. R. Finch, Mathematical Constants, Cambridge, 2003, Section 5.14., p.356.
- R. K. Guy, personal communication.
- E. Netto, Lehrbuch der Combinatorik. 2nd ed., Teubner, Leipzig, 1927, p. 96.
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Links
- G. C. Greubel, Table of n, a(n) for n = 5..1000
- R. K. Guy, Letter to N. J. A. Sloane with attachment, Mar 1988
- B. H. Margolius, Permutations with inversions, J. Integ. Seqs. Vol. 4 (2001), #01.2.4.
- R. H. Moritz and R. C. Williams, A coin-tossing problem and some related combinatorics, Math. Mag., 61 (1988), 24-29.
Programs
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Maple
f := (x,n)->product((1-x^j)/(1-x),j=1..n); seq(coeff(series(f(x,n),x,n+2),x,n-5), n=5..40); # Barbara Haas Margolius, May 31 2001
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Mathematica
Table[SeriesCoefficient[Product[(1-x^j)/(1-x),{j,1,n}],{x,0,n-5}],{n,5,25}] (* Vaclav Kotesovec, Mar 16 2014 *)
Formula
a(n) = 2^(2*n-6)/sqrt(Pi*n)*Q*(1+O(n^{-1})), where Q is a digital search tree constant, Q = 0.2887880951... (see A048651). - corrected by Vaclav Kotesovec, Mar 16 2014
Extensions
More terms, asymptotic formula from Barbara Haas Margolius (margolius(AT)math.csuohio.edu), May 31 2001
Definition clarified by Emeric Deutsch, Aug 02 2014
Comments