A005287 Number of permutations of [n] with four inversions.

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

Author

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

Keywords

• nonn
• easy

Examples

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

References

• L. 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.
• 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).
• R. 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
• R. K. Guy, Letter to N. J. A. Sloane with attachment, Mar 1988
• R. H. Moritz and R. 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).

Crossrefs

Cf. A008302, A005286, A005288.

Programs

Magma

[n*(n+1)*(n^2+n-14)/24: n in [4..50]]; // Vincenzo Librandi, Jul 17 2011

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 *)

PARI

a(n)=if(n<4,0,n*(n+1)*(n^2+n-14)/24)

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