A264078 The maximal number of standard Young tableaux without a succession v, v+1 in a row that a single partition of n can have.

1, 1, 1, 1, 2, 3, 6, 14, 30, 76, 170, 553, 1583, 5106, 14090, 41002, 164769, 603513, 2418348, 8335804, 28704417, 109618261, 466318442, 2114095511, 10276979159, 43213859606, 175668903294, 793946150358, 3490939879402, 15500974371599, 82490059523125

Offset: 0

Author

Alois P. Heinz, Nov 02 2015

Keywords

• nonn

Comments

A standard Young tableau (SYT) without a succession v, v+1 in a row is called a nonconsecutive tableau.

Examples

			a(6) = 6: partition [2,2,1,1] has 6 standard Young tableaux without a succession v, v+1 in a row, which is maximal for a partition of n=6:
15   14   14   13   13   13
26   26   25   26   25   24
3    3    3    4    4    5
4    5    6    5    6    6
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..60
• Wikipedia, Young tableau

Crossrefs

Cf. A237770, A264051.

Programs

Maple

h:= proc(l, j) option remember; `if`(l=[], 1,
      `if`(l[1]=0, h(subsop(1=[][], l), j-1), add(
      `if`(i<>j and l[i]>0 and (i=1 or l[i]>l[i-1]),
       h(subsop(i=l[i]-1, l), i), 0), i=1..nops(l))))
    end:
g:= proc(n, i, l) `if`(n=0 or i=1, h([1$n, l[]], 0),
      `if`(i<1, 0, max(g(n, i-1, l),
      `if`(i>n, 0, g(n-i, i, [i, l[]])))))
    end:
a:= n-> g(n$2, []):
seq(a(n), n=0..30);  # Alois P. Heinz, Nov 02 2015

Mathematica

h[l_, j_] := h[l, j] = If[l == {}, 1, If[l[[1]] == 0, h[ReplacePart[l, 1 -> Sequence[]], j - 1], Sum[If[i != j && l[[i]] > 0 && (i == 1 || l[[i]] > l[[i - 1]]), h[ReplacePart[l, i -> l[[i]] - 1], i], 0], {i, 1, Length[l]} ]]]; g[n_, i_, l_] := g[n, i, l] = If[n == 0 || i == 1, h[Join[Array[1 &, n], l], 0], If[i < 1, 0, Max[g[n, i - 1, l], If[i > n, 0, g[n - i, i, Join[{i}, l]]]]]]; a[n_] := g[n, n, {}];  Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Jan 22 2016, after Alois P. Heinz *)

Formula

a(n) = max { k : A264051(n,k) > 0 }.