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

0, 0, 1, 7, 39, 188, 884, 4116, 19108, 89926, 427386, 2068934, 10163358, 50888024, 258983668, 1342912608, 7079970072, 38000183102, 207309599246, 1150329076074, 6484351459090, 37143321514076, 216001121263896, 1275332898098744, 7639400455469944, 46423461664822648

Offset: 1

Author

Wouter Meeussen, Aug 04 2013

Keywords

• nonn

Comments

A (strict) 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. A225618 (weak 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], 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[inversions[#]]& /@ Tableaux[n]],{n,13}]

Extensions

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

a(19)-a(26) from Alois P. Heinz, Aug 08 2013