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A225617 Number of (strict) inversions in all standard Young tableaux of size n. 2
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 (list; graph; refs; listen; history; text; internal format)
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: A027482 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

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Last modified May 23 01:01 EDT 2017. Contains 286908 sequences.