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A214021 Number A(n,k) of n X k nonconsecutive tableaux; square array A(n,k), n>=0, k>=0, read by antidiagonals.

%I A214021 #48 Jan 22 2020 21:02:35
%S A214021 1,1,1,1,1,1,1,0,1,1,1,0,1,1,1,1,0,1,2,1,1,1,0,1,6,6,1,1,1,0,1,22,72,
%T A214021 18,1,1,1,0,1,92,1289,960,57,1,1,1,0,1,422,29889,93964,14257,186,1,1,
%U A214021 1,0,1,2074,831174,13652068,8203915,228738,622,1,1
%N A214021 Number A(n,k) of n X k nonconsecutive tableaux; square array A(n,k), n>=0, k>=0, read by antidiagonals.
%C A214021 A standard Young tableau (SYT) where entries i and i+1 never appear in the same row is called a nonconsecutive tableau.
%H A214021 Alois P. Heinz, <a href="/A214021/b214021.txt">Antidiagonals n = 0..23, flattened</a>
%H A214021 T. Y. Chow, H. Eriksson and C. K. Fan, <a href="https://doi.org/10.37236/1895">Chess tableaux</a>, Elect. J. Combin., 11 (2) (2005), #A3.
%H A214021 S. Dulucq and O. Guibert, <a href="https://doi.org/10.1016/S0012-365X(96)83009-3">Stack words, standard tableaux and Baxter permutations</a>, Disc. Math. 157 (1996), 91-106.
%H A214021 Wikipedia, <a href="https://en.wikipedia.org/wiki/Young_tableau">Young tableau</a>
%e A214021 A(2,4) = 1:
%e A214021   [1 3 5 7]
%e A214021   [2 4 6 8].
%e A214021 A(4,2) = 6:
%e A214021   [1, 5]   [1, 4]   [1, 3]   [1, 4]   [1, 3]   [1, 3]
%e A214021   [2, 6]   [2, 6]   [2, 6]   [2, 5]   [2, 5]   [2, 4]
%e A214021   [3, 7]   [3, 7]   [4, 7]   [3, 7]   [4, 7]   [5, 7]
%e A214021   [4, 8]   [5, 8]   [5, 8]   [6, 8]   [6, 8]   [6, 8].
%e A214021 Square array A(n,k) begins:
%e A214021   1, 1,  1,     1,       1,          1,              1, ...
%e A214021   1, 1,  0,     0,       0,          0,              0, ...
%e A214021   1, 1,  1,     1,       1,          1,              1, ...
%e A214021   1, 1,  2,     6,      22,         92,            422, ...
%e A214021   1, 1,  6,    72,    1289,      29889,         831174, ...
%e A214021   1, 1, 18,   960,   93964,   13652068,     2621897048, ...
%e A214021   1, 1, 57, 14257, 8203915, 8134044455, 11865331748843, ...
%p A214021 b:= proc(l, t) option remember; local n, s; n, s:= nops(l),
%p A214021        add(i, i=l); `if`(s=0, 1, add(`if`(t<>i and l[i]>
%p A214021       `if`(i=n, 0, l[i+1]), b(subsop(i=l[i]-1, l), i), 0), i=1..n))
%p A214021     end:
%p A214021 A:= (n, k)-> `if`(n<1 or k<1, 1, b([k$n], 0)):
%p A214021 seq(seq(A(n, d-n), n=0..d), d=0..12);
%t A214021 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 && l[[i]] > If[i == n, 0, l[[i+1]]], b[ReplacePart[l, i -> l[[i]]-1], i], 0], {i, 1, n}]] ] ; a[n_, k_] := If[n < 1 || k < 1, 1, b[Array[k&, n], 0]]; Table[Table[a[n, d - n], {n, 0, d}], {d, 0, 12}] // Flatten (* _Jean-François Alcover_, Dec 09 2013, translated from Maple *)
%Y A214021 Rows n=0+2, 3-4 give: A000012, A001181(k) for k>0, A214875.
%Y A214021 Columns k=0+1, 2, 3 give: A000012, A000957(n+1), A214159.
%Y A214021 Main diagonal gives A264103.
%Y A214021 Cf. A214020, A214088.
%K A214021 nonn,tabl
%O A214021 0,19
%A A214021 _Alois P. Heinz_, Jul 01 2012