A005287 Number of permutations of [n] with four inversions. (Formerly M3894)

5, 20, 49, 98, 174, 285, 440, 649, 923, 1274, 1715, 2260, 2924, 3723, 4674, 5795, 7105, 8624, 10373, 12374, 14650, 17225, 20124, 23373, 26999, 31030, 35495, 40424, 45848, 51799, 58310, 65415, 73149, 81548, 90649, 100490, 111110, 122549, 134848, 148049

Offset

4,1

References

Louis Comtet, Advanced Combinatorics, Reidel, 1974, p. 255, #2, b(n,4).

F. N. David, M. G. Kendall, and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 241.

Richard K. Guy, personal communication.

Eugen 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).

Richard P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 1, 1999; see Exercise 1.30, p. 49.

Links

Vincenzo Librandi, Table of n, a(n) for n = 4..10000

Richard K. Guy, Letter to N. J. A. Sloane with attachment, Mar 1988

Roger H. Moritz and Robert C. Williams, A coin-tossing problem and some related combinatorics, Math. Mag., 61 (1988), 24-29.

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.

Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Formula

a(n) = n*(n+1)*(n^2+n-14)/24.

G.f.: x^4*(-5 + 5*x + x^2 - 3*x^3 + x^4) / (x-1)^5. - Simon Plouffe in his 1992 dissertation

a(n) = binomial(n+1,4) + binomial(n+1,3) - binomial(n+1,2). - Zerinvary Lajos, Jul 23 2006

Sum_{n>=4} 1/a(n) = 89/98 + (4/7) * sqrt(3/19) * Pi * tan(sqrt(57)*Pi/2). - Amiram Eldar, Oct 25 2025

Example

[2, 4, 3, 1], [3, 2, 4, 1], [3, 4, 1, 2], [4, 1, 3, 2], [4, 2, 1, 3] have 4 inversions.

Maple

[seq(binomial(n, 4)+binomial(n, 3)-binomial(n, 2), n=5..43)]; # Zerinvary Lajos, Jul 23 2006

Mathematica

CoefficientList[Series[(z^4 - 3*z^3 + z^2 + 5*z - 5)/(z - 1)^5, {z, 0, 100}], z] (* Vladimir Joseph Stephan Orlovsky, Jul 16 2011 *)
LinearRecurrence[{5, -10, 10, -5, 1}, {5, 20, 49, 98, 174}, 40] (* Harvey P. Dale, Aug 25 2016 *)

Prog

(PARI) a(n)=if(n<4, 0, n*(n+1)*(n^2+n-14)/24)
(Magma) [n*(n+1)*(n^2+n-14)/24: n in [4..50]]; // Vincenzo Librandi, Jul 17 2011

Crossrefs

Cf. A008302, A005286, A005288.

Sequence in context: A168011 A160749 A147002 * A147488 A297569 A190094

Adjacent sequences: A005284 A005285 A005286 * A005288 A005289 A005290

Keyword

nonn,easy

Author

N. J. A. Sloane and Robert G. Wilson v

Status

approved