OFFSET
0,4
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]
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.
FORMULA
EXAMPLE
a(3)=4 because in the involutions 123, 132, 213, and 321 we have 0 + 1 + 1 + 2 descents.
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 *)
PROG
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Jun 09 2009
STATUS
approved