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A214461 Number of 5 X n nonconsecutive chess tableaux.

%I A214461 #26 Oct 05 2018 20:09:07
%S A214461 1,1,2,7,35,212,1421,10128,75724,593905,4927764,43884083,426665219,
%T A214461 4591883867,54943702996,725753304587,10428313247758,160361700765626,
%U A214461 2602492613187389,44111609147837212,775162479337156853,14049351313802046511,261640973700411314373
%N A214461 Number of 5 X n nonconsecutive chess tableaux.
%C A214461 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 A214461 Vaclav Kotesovec, <a href="/A214461/b214461.txt">Table of n, a(n) for n = 0..114</a> (terms 0..70 from Alois P. Heinz)
%H A214461 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 A214461 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 A214461 Wikipedia, <a href="https://en.wikipedia.org/wiki/Young_tableau">Young tableau</a>
%e A214461 a(3) = 7:
%e A214461   [1  6 11] [1  4 11] [1  6  9] [1  4  9] [1  4  7] [1  4  7] [1  4  7]
%e A214461   [2  7 12] [2  5 12] [2  7 10] [2  5 10] [2  5 10] [2  5 10] [2  5  8]
%e A214461   [3  8 13] [3  8 13] [3  8 13] [3  8 13] [3  8 13] [3  6 13] [3 10 13]
%e A214461   [4  9 14] [6  9 14] [4 11 14] [6 11 14] [6 11 14] [8 11 14] [6 11 14]
%e A214461   [5 10 15] [7 10 15] [5 12 15] [7 12 15] [9 12 15] [9 12 15] [9 12 15].
%p A214461 b:= proc(l, t) option remember; local n, s;
%p A214461        n, s:= nops(l), add(i, i=l);
%p A214461       `if`(s=0, 1, add(`if`(t<>i and irem(s+i-l[i], 2)=1 and l[i]>
%p A214461       `if`(i=n, 0, l[i+1]), b(subsop(i=l[i]-1, l), i), 0), i=1..n))
%p A214461     end:
%p A214461 a:= n-> b([n$5], 0):
%p A214461 seq(a(n), n=0..25);
%t A214461 b[l_List, t_] := b[l, t] = With[{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 -> l[[i]] - 1], i], 0], {i, 1, n}]]]; a[n_] := b[Table[n, {5}], 0]; Table[a[n], {n, 0, 25}] (* _Jean-François Alcover_, Jul 15 2017, translated from Maple *)
%Y A214461 Row n=5 of A214088.
%K A214461 nonn
%O A214461 0,3
%A A214461 _Alois P. Heinz_, Jul 18 2012