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A225617 Number of (strict) inversions in all standard Young tableaux of size n.

%I A225617 #30 Feb 18 2014 11:59:36
%S A225617 0,0,1,7,39,188,884,4116,19108,89926,427386,2068934,10163358,50888024,
%T A225617 258983668,1342912608,7079970072,38000183102,207309599246,
%U A225617 1150329076074,6484351459090,37143321514076,216001121263896,1275332898098744,7639400455469944,46423461664822648
%N A225617 Number of (strict) inversions in all standard Young tableaux of size n.
%C A225617 A (strict) inversion is a pair of cells (i,j) with i<j where j appears strictly below and strictly left of i. [_Joerg Arndt_, Feb 18 2014]
%H A225617 Alois P. Heinz, <a href="/A225617/b225617.txt">Table of n, a(n) for n = 1..50</a>
%H A225617 M. Shynar, <a href="http://www-igm.univ-mlv.fr/~fpsac/FPSAC04/ARTICLES/Shynar.pdf">On Inversions in Standard Young Tableaux</a>
%H A225617 Wikipedia, <a href="http://en.wikipedia.org/wiki/Young_tableau">Young tableau</a>
%p A225617 b:= proc(l) option remember; `if`({l[]}={0}, [1, 0],
%p A225617       add(`if`(l[j]>`if`(j=1, 0, l[j-1]), (f->f+[0, f[1]*
%p A225617       add(l[h]-l[j], h=j+1..nops(l))])
%p A225617       (b(subsop(j=l[j]-1, l))), 0), j=1..nops(l)))
%p A225617     end:
%p A225617 g:= proc(n, i, l) `if`(n=0 or i=1, b([1$n, l[]]),
%p A225617       `if`(i<1, 0, g(n, i-1, l)+
%p A225617       `if`(i>n, 0, g(n-i, i, [i, l[]]))))
%p A225617     end:
%p A225617 a:= n-> g(n$2, [])[2]:
%p A225617 seq(a(n), n=1..23);  # _Alois P. Heinz_, Aug 09 2013
%t A225617 inversions[t_?TableauQ]:= Block[{t0},t0=(First[Position[t,#1]]&) /@ Range[Max[t]]; Cases[Table[{i,j},{j,2,Max[t]},{i,j-1}],{i_,j_}/; MatchQ[t0[[i]]-t0[[j]],{_?Negative,_?Positive}]->{i,j},{2}]];
%t A225617 Table[Tr[Length[inversions[#]]& /@ Tableaux[n]],{n,13}]
%Y A225617 Cf. A225618 (weak inversions), A161125 (descent numbers).
%Y A225617 Cf. A000085 (Young tableaux with n cells).
%K A225617 nonn
%O A225617 1,4
%A A225617 _Wouter Meeussen_, Aug 04 2013
%E A225617 Terms verified and more terms added, _Joerg Arndt_, Aug 07 2013
%E A225617 a(19)-a(26) from _Alois P. Heinz_, Aug 08 2013