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A264078
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The maximal number of standard Young tableaux without a succession v, v+1 in a row that a single partition of n can have.
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3
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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
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OFFSET
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0,5
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COMMENTS
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A standard Young tableau (SYT) without a succession v, v+1 in a row is called a nonconsecutive tableau.
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LINKS
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FORMULA
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a(n) = max { k : A264051(n,k) > 0 }.
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EXAMPLE
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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
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MAPLE
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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, []):
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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