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
Keywords
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
Programs
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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
Comments