A001894 Number of permutations of {1,...,n} having n-4 inversions (n>=4). (Formerly M3484 N1416)

1, 4, 14, 49, 174, 628, 2298, 8504, 31758, 119483, 452284, 1720774, 6574987, 25214332, 96997223, 374153699, 1446677555, 5605337934, 21758936146, 84604366100, 329453055975, 1284626463105, 5015200610785, 19601107218591, 76685359017750, 300294650988857, 1176939165980809

Offset

4,2

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, 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 = 4..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-5)/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(5)=4  because we have 21345, 13245, 12435, and 12354.

Maple

f := (x, n)->product((1-x^j)/(1-x), j=1..n); seq(coeff(series(f(x, n), x, n+2), x, n-4), n=4..40); # Barbara Haas Margolius, May 31 2001

Mathematica

Table[SeriesCoefficient[Product[(1-x^j)/(1-x), {j, 1, n}], {x, 0, n-4}], {n, 4, 25}] (* Vaclav Kotesovec, Mar 16 2014 *)

Crossrefs

Cf. A008302, A048651.

Sequence in context: A211305 A316974 A278026 * A215493 A079309 A026630

Adjacent sequences: A001891 A001892 A001893 * A001895 A001896 A001897

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