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 A214459 Number of n X 3 nonconsecutive chess tableaux. 4
 1, 0, 0, 1, 1, 7, 27, 128, 640, 3351, 18313, 103404, 600538, 3571717, 21683185, 134005373, 841259885, 5355078350, 34512405410, 224908338137, 1480420941781, 9833512593113, 65860442383487, 444453988418791, 3020274890688447, 20656019108074552, 142107550142684602 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,6 COMMENTS 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. LINKS Vaclav Kotesovec, Table of n, a(n) for n = 0..200 (terms 0..70 from Alois P. Heinz) 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 FORMULA a(n) ~ c * 8^n / n^4, where c = 0.250879571... - Vaclav Kotesovec, Sep 06 2017 EXAMPLE a(5) = 7: [1 6 11] [1 4 11] [1 6 9] [1 4 9] [1 4 7] [1 4 7] [1 4 7] [2 7 12] [2 5 12] [2 7 10] [2 5 10] [2 5 10] [2 5 10] [2 5 8] [3 8 13] [3 8 13] [3 8 13] [3 8 13] [3 8 13] [3 6 13] [3 10 13] [4 9 14] [6 9 14] [4 11 14] [6 11 14] [6 11 14] [8 11 14] [6 11 14] [5 10 15] [7 10 15] [5 12 15] [7 12 15] [9 12 15] [9 12 15] [9 12 15]. 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=l[i]-1, l), i), 0), i=1..n)) end: a:= n-> b([3\$n], 0): seq(a(n), n=0..25); MATHEMATICA 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 *) CROSSREFS Column k=3 of A214088. Sequence in context: A267365 A034536 A283538 * A179597 A295209 A151496 Adjacent sequences: A214456 A214457 A214458 * A214460 A214461 A214462 KEYWORD nonn AUTHOR Alois P. Heinz, Jul 18 2012 STATUS approved

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