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A225618 Number of weak inversions in all standard Young tableaux of size n. 2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

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]

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]+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}]

CROSSREFS

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

Cf. A000085 (Young tableaux with n cells).

Sequence in context: A267774 A243474 A111644 * A081278 A054146 A172062

Adjacent sequences:  A225615 A225616 A225617 * A225619 A225620 A225621

KEYWORD

nonn

AUTHOR

Wouter Meeussen, Aug 04 2013

EXTENSIONS

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

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

STATUS

approved

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Last modified March 27 22:02 EDT 2017. Contains 284182 sequences.