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

0,5

Comments

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

Links

Alois P. Heinz, Table of n, a(n) for n = 0..60

Wikipedia, Young tableau

Formula

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

Example

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

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 *)

Crossrefs

Cf. A237770, A264051.

Sequence in context: A331684 A106364 A211931 * A006444 A335242 A032047

Adjacent sequences: A264075 A264076 A264077 * A264079 A264080 A264081

Keyword

nonn

Author

Alois P. Heinz, Nov 02 2015

Status

approved