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A225618 Number of weak inversions in all standard Young tableaux of size n.

%I A225618 #31 Feb 18 2014 12:00:07
%S A225618 0,1,6,29,125,538,2282,9916,43416,195206,891638,4176002,19920914,
%T A225618 97248184,483752596,2458123328,12722535412,67155870194,360792258750,
%U A225618 1974047659038,10983669569446,62162472053580,357454683655920,2088497013864312,12387836332741800
%N A225618 Number of weak inversions in all standard Young tableaux of size n.
%C A225618 A weak inversion is a pair of cells (i,j) with i<j where j appears weakly below and weakly left of i. [_Joerg Arndt_, Feb 18 2014]
%H A225618 Alois P. Heinz, <a href="/A225618/b225618.txt">Table of n, a(n) for n = 1..50</a>
%H A225618 M. Shynar, <a href="http://www-igm.univ-mlv.fr/~fpsac/FPSAC04/ARTICLES/Shynar.pdf">On Inversions in Standard Young Tableaux</a>
%H A225618 Wikipedia, <a href="http://en.wikipedia.org/wiki/Young_tableau">Young tableau</a>
%p A225618 b:= proc(l) option remember; `if`({l[]}={0}, [1, 0],
%p A225618       add(`if`(l[j]>`if`(j=1, 0, l[j-1]), (f->f+[0, f[1]*
%p A225618       add(l[h]-l[j]+1, h=j+1..nops(l))])
%p A225618       (b(subsop(j=l[j]-1, l))), 0), j=1..nops(l)))
%p A225618     end:
%p A225618 g:= proc(n, i, l) `if`(n=0 or i=1, b([1$n, l[]]),
%p A225618       `if`(i<1, 0, g(n, i-1, l)+
%p A225618       `if`(i>n, 0, g(n-i, i, [i, l[]]))))
%p A225618     end:
%p A225618 a:= n-> g(n$2, [])[2]:
%p A225618 seq(a(n), n=1..23);  # _Alois P. Heinz_, Aug 09 2013
%t A225618 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 A225618 Table[Tr[Length[weakinversions[#]]& /@ Tableaux[n]],{n,12}]
%Y A225618 Cf. A225617 (strict inversions), A161125 (descent numbers).
%Y A225618 Cf. A000085 (Young tableaux with n cells).
%K A225618 nonn
%O A225618 1,3
%A A225618 _Wouter Meeussen_, Aug 04 2013
%E A225618 Terms verified and more terms added, _Joerg Arndt_, Aug 07 2013
%E A225618 a(19)-a(25) from _Alois P. Heinz_, Aug 08 2013