A005283 Number of permutations of (1,...,n) having n-5 inversions (n>=5). (Formerly M3905)

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

5,2

Comments

Sequence is a diagonal of the triangle A008302 (number of permutations of (1,...,n) with k inversions; see Table 1 of the Margolius reference). - Emeric Deutsch, Aug 02 2014

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.

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

Example

a(6)=5 because we have 213456, 132456, 124356, 123546, and 123465.

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

Mathematica

Table[SeriesCoefficient[Product[(1-x^j)/(1-x), {j, 1, n}], {x, 0, n-5}], {n, 5, 25}] (* Vaclav Kotesovec, Mar 16 2014 *)

Crossrefs

Cf. A008302, A000707, A001892, A001893, A001894, A005284, A005285, A048651.

Sequence in context: A270023 A061278 A000758 * A057552 A300918 A269708

Adjacent sequences: A005280 A005281 A005282 * A005284 A005285 A005286

Keyword

nonn

Author

N. J. A. Sloane

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

Status

approved