|
| |
|
|
A005283
|
|
Number of permutations of (1,...,n) having n-5 inversions (n>=5).
(Formerly M3905)
|
|
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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
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
|
| |
|
|
