0,4

a(n)=n!n(n-1)/4 gives the total number of inversions in all the permutations of [n]. [Stern, Terquem] Proof: For fixed i,j and for fixed I,J (i<j, I>J, 1 <= i,j,I,J <= n), we have (n-2)! permutations p of [n] for which p(i)=I and p(j)=J (permute {1,2,...,n} \ {I,J} in the positions (1,2,...,n) \ {i,j}). There are n(n-1)/2 choices for the pair (i,j) with i<j and n(n-1)/2 choices for the pair (I,J) with I>J. Consequently, the total number of inversions in all the permutations of [n] is (n-2)![n(n-1)/2]^2 = n!n(n-1)/4. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Oct 05 2006

Also coefficients of Laguerre polynomials. a(n)=A021009(n,2), n >= 2.

a(n) is the number of edges in the Cayley graph of the symmetric group S_n with respect to the generating set consisting of transpositions - Avi Peretz (njk(AT)netvision.net.il), Feb 20 2001

a(n+1) is the sum of the moments over all permutations of n. E.g. a(4) is [1,2,3].[1,2,3] + [1,3,2].[1,2,3] + [2,1,3].[1,2,3] + [2,3,1].[1,2,3] + [3,1,2].[1,2,3] + [3,2,1].[1,2.3] = 14 + 13 + 13 + 11 + 11 + 10 = 72. - Jon Perry (perry(AT)globalnet.co.uk), Feb 20 2004

Mahonian weight function. - Roger Bagula (rlbagulatftn(AT)yahoo.com), Oct 05 2006

Derivative of the q-factorial [n]!, evaluated at q=1. Example: a(3)=9 because (d/dq)[3]!=(d/dq)((1+q)(1+q+q^2))=2+4q+3q^2 is equal to 9 at q=1. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 19 2007

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 799.

S. Altmann and E. L. Ortiz, Eds., Mathematical and Social Utopias in France: Olinde Rodrigues and His Times, Amer. Math. Soc., 2005.

D. M. Bressoud, Proofs and Confirmations, Camb. Univ. Press, 1999; p. 90.

W. P. Johnson, Review of Altmann-Ortiz book, Amer. Math. Monthly, 114 (2007), 752-758.

C. Lanczos, Applied Analysis. Prentice-Hall, Englewood Cliffs, NJ, 1956, p. 519.

E. Reingold, J. Nievergelt and N. Deo, Combinatorial Algorithms, Prentice-Hall, 1977, section 7.1, p. 287.

M. P. Schutzenberger and D. Foata, Major index and inversion numbers of permutations, Math. Nachr., t. 83, 1978, p. 143-159.

M. Stern, Crelle, 18 (1838), p. 100.

O. Terquem, Liouville's Journal, 1838.

Thanatipanonda, Thotsaporn, Inversions and major index for permutations, Math. Mag., April 2004.

T. D. Noe, Table of n, a(n) for n=0..100

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

Eric Babson, Einar Steingrimsson, Generalized Permutaion Patterns and Classification of the Mahonian statistics

Index entries for sequences related to Laguerre polynomials

E.g.f.: (1/2)*x^2/(1-x)^3. a(n)=a(n-1)*n^2/(n-2), n>2. a(2)=1.

a(n) = n*a(n-1)+(n-1)!*n*(n-1)/2, a(1) = 0, a(2) = 1; a(n) = sum (n! first terms of A034968); a(n) = sum of the rises j of permutations (p(j)<p(j+1)) - Claude Lenormand (claude.lenormand(AT)free.fr), Feb 02 2001

If we define f(n,i,x)= sum(sum(binomial(k,j)*stirling1(n,k)*stirling2(j,i)*x^(k-j),j=i..k),k=i..n) then a(n)=(-1)^n*f(n,2,-3), (n>=2). [From Milan R. Janjic (agnus(AT)blic.net), Mar 01 2009]

A001809 := n->n!*n*(n-1)/4;

with(combstruct):ZL:=[st, {st=Prod(left, right), left=Set(U, card=r), right=Set(U, card<r), U=Sequence(Z, card>=1)}, labeled]: subs(r=1, stack): seq(count(subs(r=2, ZL), size=m), m=0..19) ; - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Feb 07 2008

with (combstruct):with (combinat):a:=proc(m) [ZL, {ZL=Set(Cycle(Z, card>=m))}, labeled]; end: ZLL:=a(1):seq(count(ZLL, size=n)*binomial(n, 2)/2, n=0..21); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 11 2008

(PARI) a(n)=n!*n*(n-1)/4

sage: [factorial(m)*binomial(m, 2)/2 for m in xrange (0, 19)] - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 05 2008

Cf. A034968 (the inversion numbers of permutations listed in alphabetic order). See also A053495 and A064038.

Cf. A008302.

Sequence in context: A003951 A033135 A127053 this_sequence A006135 A133672 A043079

Adjacent sequences: A001806 A001807 A001808 this_sequence A001810 A001811 A001812

nonn,easy,nice,eigen

N. J. A. Sloane (njas(AT)research.att.com).

More terms and new description from Michael Somos, May 19, 2000.

Simpler description from Emeric Deutsch, Oct 05 2006