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A214459 Number of n X 3 nonconsecutive chess tableaux.

%I A214459 #32 Oct 05 2018 20:08:37
%S A214459 1,0,0,1,1,7,27,128,640,3351,18313,103404,600538,3571717,21683185,
%T A214459 134005373,841259885,5355078350,34512405410,224908338137,
%U A214459 1480420941781,9833512593113,65860442383487,444453988418791,3020274890688447,20656019108074552,142107550142684602
%N A214459 Number of n X 3 nonconsecutive chess tableaux.
%C A214459 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 A214459 Vaclav Kotesovec, <a href="/A214459/b214459.txt">Table of n, a(n) for n = 0..200</a> (terms 0..70 from Alois P. Heinz)
%H A214459 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 A214459 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 A214459 Wikipedia, <a href="https://en.wikipedia.org/wiki/Young_tableau">Young tableau</a>
%F A214459 a(n) ~ c * 8^n / n^4, where c = 0.250879571... - _Vaclav Kotesovec_, Sep 06 2017
%e A214459 a(5) = 7:
%e A214459   [1  6 11] [1  4 11] [1  6  9] [1  4  9] [1  4  7] [1  4  7] [1  4  7]
%e A214459   [2  7 12] [2  5 12] [2  7 10] [2  5 10] [2  5 10] [2  5 10] [2  5  8]
%e A214459   [3  8 13] [3  8 13] [3  8 13] [3  8 13] [3  8 13] [3  6 13] [3 10 13]
%e A214459   [4  9 14] [6  9 14] [4 11 14] [6 11 14] [6 11 14] [8 11 14] [6 11 14]
%e A214459   [5 10 15] [7 10 15] [5 12 15] [7 12 15] [9 12 15] [9 12 15] [9 12 15].
%p A214459 b:= proc(l, t) option remember; local n, s;
%p A214459        n, s:= nops(l), add(i, i=l);
%p A214459       `if`(s=0, 1, add(`if`(t<>i and irem(s+i-l[i], 2)=1 and l[i]>
%p A214459       `if`(i=n, 0, l[i+1]), b(subsop(i=l[i]-1, l), i), 0), i=1..n))
%p A214459     end:
%p A214459 a:= n-> b([3$n], 0):
%p A214459 seq(a(n), n=0..25);
%t A214459 b[l_, t_] := b[l, t] = Module[{n, s}, {n, s} = {Length[l], Sum[i, {i, 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 -> l[[i]] - 1}], i], 0], {i, 1, n}]]]; a[n_] := If[n < 1, 1, b[Array[3&, n], 0]]; Table[a[n], {n, 0, 26}] (* _Jean-François Alcover_, Jul 13 2017, after _Alois P. Heinz_ *)
%Y A214459 Column k=3 of A214088.
%K A214459 nonn
%O A214459 0,6
%A A214459 _Alois P. Heinz_, Jul 18 2012