A238020 Number of nonconsecutive chess tableaux with n cells.
1, 1, 1, 1, 2, 2, 4, 5, 10, 15, 33, 52, 126, 213, 537, 991, 2563, 5118, 13670, 29171, 81069, 180813, 525755, 1216996, 3693934, 8843831, 27797975, 69106326, 223116931, 577433770, 1903516721, 5136516772, 17257698892, 48388514996, 166022450140, 481137194184
Offset: 0
Keywords
Examples
a(6) = 4: [1] [1 6] [1 4] [1 4] [2] [2] [2 5] [2 5] [3] [3] [3] [3 6] [4] [4] [6] [5] [5] [6]
Links
- Alois P. Heinz, Table of n, a(n) for n = 0..50
- T. Y. Chow, H. Eriksson and C. K. Fan, Chess tableaux, Elect. J. Combin., 11 (2) (2005), #A3.
- Jonas Sjöstrand, On the sign-imbalance of partition shapes, arXiv:math/0309231v3 [math.CO], 2005.
- Wikipedia, Young tableau
Programs
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Maple
b:= proc(l, t) option remember; local n, s; n, s:= nops(l), add(i, i=l); `if`(s=0, 1, add(`if`(t<>i and irem(s+i-l[i], 2)=1 and l[i]>`if`(i=n, 0, l[i+1]), b(subsop( i=`if`(i=n and l[n]=1, [][], l[i]-1), l), i), 0), i=1..n)) end: g:= (n, i, l)-> `if`(n=0 or i=1, b([l[], 1$n], 0), `if`(i<1, 0, add(g(n-i*j, i-1, [l[], i$j]), j=0..n/i))): a:= n-> g(n, n, []): seq(a(n), n=0..32); -
Mathematica
b[l_, t_] := b[l, t] = Module[{ n = Length[l], s = Total[l]}, If[s == 0, 1, Sum[If[t != i && Mod[s + i - l[[i]], 2] == 1 && l[[i]] > If[i == n, 0, l[[i + 1]]], b[ReplacePart[l, i -> If[i == n && l[[n]] == 1, Nothing, l[[i]] - 1]], i], 0], {i, 1, n}]]]; g[n_, i_, l_] := If[n == 0 || i == 1, b[Join[l, Table[1, n]], 0], If[i < 1, 0, Sum[g[n - i*j, i - 1, Join[l, Table[i, j]]], {j, 0, n/i}]]]; a[n_] := g[n, n, {}]; Table[a[n], {n, 0, 32}] (* Jean-François Alcover, Nov 08 2017, after Alois P. Heinz *)
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