cp's OEIS Frontend

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.

%I A264078 #18 Feb 06 2017 14:39:02
%S A264078 1,1,1,1,2,3,6,14,30,76,170,553,1583,5106,14090,41002,164769,603513,
%T A264078 2418348,8335804,28704417,109618261,466318442,2114095511,10276979159,
%U A264078 43213859606,175668903294,793946150358,3490939879402,15500974371599,82490059523125
%N 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.
%C A264078 A standard Young tableau (SYT) without a succession v, v+1 in a row is called a nonconsecutive tableau.
%H A264078 Alois P. Heinz, <a href="/A264078/b264078.txt">Table of n, a(n) for n = 0..60</a>
%H A264078 Wikipedia, <a href="https://en.wikipedia.org/wiki/Young_tableau">Young tableau</a>
%F A264078 a(n) = max { k : A264051(n,k) > 0 }.
%e A264078 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:
%e A264078 15   14   14   13   13   13
%e A264078 26   26   25   26   25   24
%e A264078 3    3    3    4    4    5
%e A264078 4    5    6    5    6    6
%p A264078 h:= proc(l, j) option remember; `if`(l=[], 1,
%p A264078       `if`(l[1]=0, h(subsop(1=[][], l), j-1), add(
%p A264078       `if`(i<>j and l[i]>0 and (i=1 or l[i]>l[i-1]),
%p A264078        h(subsop(i=l[i]-1, l), i), 0), i=1..nops(l))))
%p A264078     end:
%p A264078 g:= proc(n, i, l) `if`(n=0 or i=1, h([1$n, l[]], 0),
%p A264078       `if`(i<1, 0, max(g(n, i-1, l),
%p A264078       `if`(i>n, 0, g(n-i, i, [i, l[]])))))
%p A264078     end:
%p A264078 a:= n-> g(n$2, []):
%p A264078 seq(a(n), n=0..30);  # _Alois P. Heinz_, Nov 02 2015
%t A264078 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_ *)
%Y A264078 Cf. A237770, A264051.
%K A264078 nonn
%O A264078 0,5
%A A264078 _Alois P. Heinz_, Nov 02 2015