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

0, 1, 6, 29, 125, 538, 2282, 9916, 43416, 195206, 891638, 4176002, 19920914, 97248184, 483752596, 2458123328, 12722535412, 67155870194, 360792258750, 1974047659038, 10983669569446, 62162472053580, 357454683655920, 2088497013864312, 12387836332741800

Offset: 1

Author

Wouter Meeussen, Aug 04 2013

Keywords

• nonn

Comments

A weak inversion is a pair of cells (i,j) with iJoerg Arndt, Feb 18 2014]

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..50
• M. Shynar, On Inversions in Standard Young Tableaux
• Wikipedia, Young tableau

Crossrefs

Cf. A225617 (strict inversions), A161125 (descent numbers).

Cf. A000085 (Young tableaux with n cells).

Programs

Maple

b:= proc(l) option remember; `if`({l[]}={0}, [1, 0],
      add(`if`(l[j]>`if`(j=1, 0, l[j-1]), (f->f+[0, f[1]*
      add(l[h]-l[j]+1, h=j+1..nops(l))])
      (b(subsop(j=l[j]-1, l))), 0), j=1..nops(l)))
    end:
g:= proc(n, i, l) `if`(n=0 or i=1, b([1$n, l[]]),
      `if`(i<1, 0, g(n, i-1, l)+
      `if`(i>n, 0, g(n-i, i, [i, l[]]))))
    end:
a:= n-> g(n$2, [])[2]:
seq(a(n), n=1..23);  # Alois P. Heinz, Aug 09 2013

Mathematica

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}]];
Table[Tr[Length[weakinversions[#]]& /@ Tableaux[n]],{n,12}]

Extensions

Terms verified and more terms added, Joerg Arndt, Aug 07 2013

a(19)-a(25) from Alois P. Heinz, Aug 08 2013