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A238020 Number of nonconsecutive chess tableaux with n cells.

%I A238020 #22 Nov 08 2017 10:26:52
%S A238020 1,1,1,1,2,2,4,5,10,15,33,52,126,213,537,991,2563,5118,13670,29171,
%T A238020 81069,180813,525755,1216996,3693934,8843831,27797975,69106326,
%U A238020 223116931,577433770,1903516721,5136516772,17257698892,48388514996,166022450140,481137194184
%N A238020 Number of nonconsecutive chess tableaux with n cells.
%C A238020 A standard Young tableau (SYT) with cell(i,j)+i+j == 1 mod 2 for all cells where entries m and m+1 never appear in the same row is called a nonconsecutive chess tableau.
%H A238020 Alois P. Heinz, <a href="/A238020/b238020.txt">Table of n, a(n) for n = 0..50</a>
%H A238020 T. Y. Chow, H. Eriksson and C. K. Fan, <a href="http://www.combinatorics.org/Volume_11/Abstracts/v11i2a3.html">Chess tableaux</a>, Elect. J. Combin., 11 (2) (2005), #A3.
%H A238020 Jonas Sjöstrand, <a href="https://arxiv.org/abs/math/0309231v3">On the sign-imbalance of partition shapes</a>, arXiv:math/0309231v3 [math.CO], 2005.
%H A238020 Wikipedia, <a href="https://en.wikipedia.org/wiki/Young_tableau">Young tableau</a>
%e A238020 a(6) = 4:
%e A238020 [1]   [1 6]   [1 4]   [1 4]
%e A238020 [2]   [2]     [2 5]   [2 5]
%e A238020 [3]   [3]     [3]     [3 6]
%e A238020 [4]   [4]     [6]
%e A238020 [5]   [5]
%e A238020 [6]
%p A238020 b:= proc(l, t) option remember; local n, s;
%p A238020       n, s:= nops(l), add(i, i=l); `if`(s=0, 1, add(`if`(t<>i and
%p A238020       irem(s+i-l[i], 2)=1 and l[i]>`if`(i=n, 0, l[i+1]), b(subsop(
%p A238020       i=`if`(i=n and l[n]=1, [][], l[i]-1), l), i), 0), i=1..n))
%p A238020     end:
%p A238020 g:= (n, i, l)-> `if`(n=0 or i=1, b([l[], 1$n], 0), `if`(i<1, 0,
%p A238020                  add(g(n-i*j, i-1, [l[], i$j]), j=0..n/i))):
%p A238020 a:= n-> g(n, n, []):
%p A238020 seq(a(n), n=0..32);
%t A238020 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}]]];
%t A238020 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}]]];
%t A238020 a[n_] := g[n, n, {}];
%t A238020 Table[a[n], {n, 0, 32}] (* _Jean-François Alcover_, Nov 08 2017, after _Alois P. Heinz_ *)
%Y A238020 Cf. A214088, A214459, A214460, A214461, A237770, A238014, A238184.
%K A238020 nonn
%O A238020 0,5
%A A238020 _Alois P. Heinz_, Feb 17 2014