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

1,4

Comments

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]

Links

Alois P. Heinz, Table of n, a(n) for n = 1..50

M. Shynar, On Inversions in Standard Young Tableaux

Wikipedia, Young tableau

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}]

Crossrefs

Cf. A225618 (weak inversions), A161125 (descent numbers).

Cf. A000085 (Young tableaux with n cells).

Sequence in context: A352310 A196782 A128554 * A032207 A004057 A081898

Adjacent sequences: A225614 A225615 A225616 * A225618 A225619 A225620

Keyword

nonn

Author

Wouter Meeussen, Aug 04 2013

Extensions

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

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

Status

approved