A090672 a(n) = (n^2-1)*n!/3.

0, 2, 16, 120, 960, 8400, 80640, 846720, 9676800, 119750400, 1596672000, 22832409600, 348713164800, 5666588928000, 97639686144000, 1778437140480000, 34145993097216000, 689322235650048000, 14597412049059840000, 323575967087493120000, 7493338185184051200000

Offset: 1

Author

N. J. A. Sloane, Dec 18 2003

Keywords

• nonn

Comments

a(n) = Sum_{pi in Symm(n)} Sum_{i=1..n} |pi(i)-i|, i.e., the total displacement of all letters in all permutations on n letters.

a(n) = number of entries between the entries 1 and 2 in all permutations of {1,2,...,n+1}. Example: a(2)=2 because we have 123, 1(3)2, 213, 2(3)1, 312, 321; the entries between 1 and 2 are surrounded by parentheses. - Emeric Deutsch, Apr 06 2008

a(n) = Sum_{k=0..n-1} k*A138770(n+1,k). - Emeric Deutsch, Apr 06 2008

a(n) is also the number of peaks in all permutations of {1,2,...,n+1}. Example: a(3)=16 because the permutations 1234, 4123, 3124, 4312, 2134, 4213, 3214, and 4321 have no peaks and each of the remaining 16 permutations of {1,2,3,4} has exactly one peak. - Emeric Deutsch, Jul 26 2009

a(n), for n>=2, is the number of (n+2)-node tournaments that have exactly one triad. Proven by Kadane (1966), see link. - Ian R Harris, Sep 26 2022

References

• D. Daly and P. Vojtechovsky, Displacement of permutations, preprint, 2003.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..300
• Ian R. Harris and Ryan P. A. McShane, Counting Tournaments with a Specified Number of Circular Triads, Journal of Integer Sequences, Vol. 27 (2024), Article 24.8.7. See pages 2, 23.
• J. B. Kadane, Some equivalence classes in paired comparisons, The Annals of Mathematical Statistics, 37 (1966), 488-494.

Crossrefs

Twice A005990.

Cf. A138770.

Cf. A001113, A001620, A091725, A099285.

Programs

Magma

[(n^2-1)*Factorial(n)/3: n in [1..25]]; // Vincenzo Librandi, Oct 11 2011

Mathematica

nn=20;Drop[Range[0,nn]!CoefficientList[Series[ x^3/3/(1-x)^2,{x,0,nn}],x],2]  (* Geoffrey Critzer, Mar 04 2013 *)

Formula

a(n) = A052571(n+2)/3 = 2*A005990(n). - Zerinvary Lajos, May 11 2007

a(n) = (n+3)! * Sum_{k=1..n} (k+1)!/(k+3)!, with offset 0. - Gary Detlefs, Aug 05 2010

E.g.f.: (x^3 - 3*x^2)/(3*(x-1)^3). - Geoffrey Critzer, Mar 04 2013

From Amiram Eldar, May 14 2022: (Start)

Sum_{n>=2} 1/a(n) = (3/2)*(Ei(1) - gamma) - 3*e + 27/4, where Ei(1) = A091725, gamma = A001620, and e = A001113.

Sum_{n>=2} (-1)^n/a(n) = (3/2)*(gamma - Ei(-1)) - 3/4, where Ei(-1) = -A099285. (End)