A161125 Number of descents in all involutions of {1,2,...,n}.

0, 0, 1, 4, 15, 52, 190, 696, 2674, 10480, 42732, 178480, 770836, 3411024, 15538120, 72446752, 346550520, 1694394496, 8477167504, 43287312960, 225707308912, 1199526928960, 6498288708576, 35836282708864, 201160191642400, 1148165325126912, 6662315102507200, 39268797697682176

Offset: 0

Author

Emeric Deutsch, Jun 09 2009

Keywords

• nonn

Comments

Also total number of descents in all tableaux of size n (see Stanley ref.).

A descent in a standard Young tableau is a entry i such that i+1 lies strictly below and weakly left of i. [Joerg Arndt, Feb 18 2014]

Examples

			a(3)=4 because in the involutions 123, 132, 213, and 321 we have 0 + 1 + 1 + 2 descents.
		

References

• R. P. Stanley, Enumerative Combinatorics Vol 2., Lemma 7.19.6, p. 361

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..500
• J. Désarménien and D. Foata, Fonctions symétriques et séries hypergéométriques basiques multivariées, Bull. Soc. Math. France, 113, 1985, 3-22.
• I. M. Gessel and C. Reutenauer, Counting permutations with given cycle structure and descent set, J. Combin. Theory, Ser. A, 64, 1993, 189-215.
• V. J. W. Guo and J. Zeng, The Eulerian distribution on involutions is indeed unimodal, J. Combin. Theory, Ser. A, 113, 2006, 1061-1071.

Crossrefs

Cf. A000085, A161126.

Programs

Maple

a[0] := 0: a[1] := 0: a[2] := 1: a[3] := 4: for n from 4 to 27 do a[n] := (n-1)*(a[n-1]/(n-2)+(n-1)*a[n-2]/(n-3)) end do: seq(a[n], n = 0 .. 27); # end of program
g := (1-(1-z-z^2)*exp(z+(1/2)*z^2))*1/2: gser := series(g, z = 0, 30): seq(factorial(n)*coeff(gser, z, n), n = 0 .. 27); # end of program

Mathematica

CoefficientList[Series[(1-(1-z-z^2)*Exp[z+(1/2)*z^2])/2,{z,0,24}],z] Range[0,24]!; (* Emeric Deutsch, Jun 09 2009 *)
descentset[t_?TableauQ]:=Sort[Cases[t,i_Integer /; Position[t,i+1][[1,1]] > Position[t,i][[1,1]], {2}]]; Table[Tr[Length[descentset[#]]& /@Tableaux[n]], {n,1,12}] (* Wouter Meeussen, Aug 04 2013 *)

PARI

x='x+O('x^66);  concat([0,0],Vec(serlaplace((1/2)*(1-(1-x-x^2)*exp(x+x^2/2))))) \\ Joerg Arndt, Aug 04 2013

Formula

a(n) = (n-1)*A000085(n)/2.

a(n) = Sum(k*A161126(n,k), k=0..n-1).

Rec. rel.: a(n)/(n-1) = a(n-1)/(n-2) + (n-1)*a(n-2)/(n-3) for n>=4 (see 1st Maple program).

E.g.f.: (1/2)*(1 - (1 - z - z^2)*exp(z + z^2/2)).