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A161125
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Number of descents in all involutions of {1,2,...,n}.
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4
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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
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OFFSET
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0,4
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COMMENTS
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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]
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REFERENCES
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R. P. Stanley, Enumerative Combinatorics Vol 2., Lemma 7.19.6, p. 361
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LINKS
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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.
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FORMULA
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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)).
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EXAMPLE
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a(3)=4 because in the involutions 123, 132, 213, and 321 we have 0 + 1 + 1 + 2 descents.
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MAPLE
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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
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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Cf. A000085, A161126.
Sequence in context: A117917 A192431 A329253 * A027295 A208722 A057332
Adjacent sequences: A161122 A161123 A161124 * A161126 A161127 A161128
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KEYWORD
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nonn
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AUTHOR
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Emeric Deutsch, Jun 09 2009
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STATUS
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approved
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