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

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, 18, 1, 1, 1, 0, 1, 92, 1289, 960, 57, 1, 1, 1, 0, 1, 422, 29889, 93964, 14257, 186, 1, 1, 1, 0, 1, 2074, 831174, 13652068, 8203915, 228738, 622, 1, 1

Offset: 0

Author

Alois P. Heinz, Jul 01 2012

Keywords

• nonn
• tabl

Comments

A standard Young tableau (SYT) where entries i and i+1 never appear in the same row is called a nonconsecutive tableau.

Examples

			A(2,4) = 1:
  [1 3 5 7]
  [2 4 6 8].
A(4,2) = 6:
  [1, 5]   [1, 4]   [1, 3]   [1, 4]   [1, 3]   [1, 3]
  [2, 6]   [2, 6]   [2, 6]   [2, 5]   [2, 5]   [2, 4]
  [3, 7]   [3, 7]   [4, 7]   [3, 7]   [4, 7]   [5, 7]
  [4, 8]   [5, 8]   [5, 8]   [6, 8]   [6, 8]   [6, 8].
Square array A(n,k) begins:
  1, 1,  1,     1,       1,          1,              1, ...
  1, 1,  0,     0,       0,          0,              0, ...
  1, 1,  1,     1,       1,          1,              1, ...
  1, 1,  2,     6,      22,         92,            422, ...
  1, 1,  6,    72,    1289,      29889,         831174, ...
  1, 1, 18,   960,   93964,   13652068,     2621897048, ...
  1, 1, 57, 14257, 8203915, 8134044455, 11865331748843, ...
		

Links

• Alois P. Heinz, Antidiagonals n = 0..23, flattened
• T. Y. Chow, H. Eriksson and C. K. Fan, Chess tableaux, Elect. J. Combin., 11 (2) (2005), #A3.
• S. Dulucq and O. Guibert, Stack words, standard tableaux and Baxter permutations, Disc. Math. 157 (1996), 91-106.
• Wikipedia, Young tableau

Crossrefs

Rows n=0+2, 3-4 give: A000012, A001181(k) for k>0, A214875.

Columns k=0+1, 2, 3 give: A000012, A000957(n+1), A214159.

Main diagonal gives A264103.

Cf. A214020, A214088.

Programs

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 l[i]>
      `if`(i=n, 0, l[i+1]), b(subsop(i=l[i]-1, l), i), 0), i=1..n))
    end:
A:= (n, k)-> `if`(n<1 or k<1, 1, b([k$n], 0)):
seq(seq(A(n, d-n), n=0..d), d=0..12);

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 && 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 *)