A001892 Number of permutations of (1,...,n) having n-2 inversions (n>=2). (Formerly M1477 N0583)

1, 2, 5, 15, 49, 169, 602, 2191, 8095, 30239, 113906, 431886, 1646177, 6301715, 24210652, 93299841, 360490592, 1396030396, 5417028610, 21056764914, 81978913225, 319610939055, 1247641114021, 4875896455975, 19075294462185, 74696636715792, 292758662041150

Offset

2,2

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.

E. Netto, Lehrbuch der Combinatorik. 2nd ed., Teubner, Leipzig, 1927, p. 96.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

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 = 2..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.

E. Netto, Lehrbuch der Combinatorik, Chapter 4, annotated scanned copy of pages 92-99 only.

Formula

a(n) = 2^(2*n-3)/sqrt(Pi*n)*Q*(1+O(n^{-1})), where Q is a digital search tree constant, Q = 0.288788095... (see A048651). - corrected by Vaclav Kotesovec, Mar 16 2014

Example

a(4)=5  because we have 1342, 1423, 2143, 2314, and 3124.

Maple

f := (x, n)->product((1-x^j)/(1-x), j=1..n); seq(coeff(series(f(x, n), x, n+2), x, n-2), n=2..40);

Mathematica

Table[SeriesCoefficient[Product[(1-x^j)/(1-x), {j, 1, n}], {x, 0, n-2}], {n, 2, 25}] (* Vaclav Kotesovec, Mar 16 2014 *)

Crossrefs

Cf. A008302, A048651.

Sequence in context: A149937 A149938 A148365 * A176025 A084082 A190270

Adjacent sequences: A001889 A001890 A001891 * A001893 A001894 A001895

Keyword

nonn

Author

N. J. A. Sloane, R. K. Guy

Extensions

More terms, Maple code, asymptotic formula from Barbara Haas Margolius (margolius(AT)math.csuohio.edu), May 31 2001

Definition clarified by Emeric Deutsch, Aug 02 2014

Status

approved