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A238802 Number T(n,k) of standard Young tableaux with n cells where k is the length of the maximal consecutive sequence 1,2,...,k in the first column; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

%I A238802 #37 Feb 06 2017 17:57:06
%S A238802 1,0,1,0,1,1,0,2,1,1,0,5,3,1,1,0,13,8,3,1,1,0,38,24,9,3,1,1,0,116,74,
%T A238802 28,9,3,1,1,0,382,246,93,29,9,3,1,1,0,1310,848,321,98,29,9,3,1,1,0,
%U A238802 4748,3088,1168,350,99,29,9,3,1,1,0,17848,11644,4404,1302,356,99,29,9,3,1,1
%N A238802 Number T(n,k) of standard Young tableaux with n cells where k is the length of the maximal consecutive sequence 1,2,...,k in the first column; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
%C A238802 T(0,0) = 1 by convention.
%C A238802 Also the number of ballot sequences of length n with exactly k fixed points.  The fixed points are in the positions 1,2,...,k.
%C A238802 Row sums give A000085.
%C A238802 Diagonal T(2n,n) gives A238803(n).
%C A238802 Diagonal T(2n+1,n) gives A238803(n+1)-1.
%C A238802 T(n,1) = Sum_{k=2..n} T(n,k) = A000085(n)/2 = A001475(n-1) for n>1.
%C A238802 Columns k=2-8 give: A238977, A238978, A238979, A239116, A239117, A239118, A239119.
%C A238802 Conjecture: Generally, column k is asymptotic to sqrt(2)/(2*(k+1)*(k-1)!) * exp(sqrt(n)-n/2-1/4) * n^(n/2) * (1 + 7/(24*sqrt(n))), holds for all k<=10. - _Vaclav Kotesovec_, Mar 08 2014
%H A238802 Joerg Arndt and Alois P. Heinz, <a href="/A238802/b238802.txt">Rows n = 0..50, flattened</a>
%H A238802 Wikipedia, <a href="https://en.wikipedia.org/wiki/Young_tableau">Young tableau</a>
%e A238802 The 10 tableaux with n=4 cells sorted by the length of the maximal consecutive sequence 1,2,...,k in the first column are:
%e A238802 :[1 2] [1 2] [1 2 3] [1 2 4] [1 2 3 4]:[1 3] [1 3] [1 3 4]:[1 4]:[1]:
%e A238802 :[3]   [3 4] [4]     [3]              :[2]   [2 4] [2]    :[2]  :[2]:
%e A238802 :[4]                                  :[4]                :[3]  :[3]:
%e A238802 :                                     :                   :     :[4]:
%e A238802 : -----------------1----------------- : --------2-------- : -3- : 4 :
%e A238802 Their corresponding ballot sequences are:
%e A238802 [1, 1, 2, 3]  ->  1 \
%e A238802 [1, 1, 2, 2]  ->  1  \
%e A238802 [1, 1, 1, 2]  ->  1   } -- 5
%e A238802 [1, 1, 2, 1]  ->  1  /
%e A238802 [1, 1, 1, 1]  ->  1 /
%e A238802 [1, 2, 1, 3]  ->  2 \
%e A238802 [1, 2, 1, 2]  ->  2  } --- 3
%e A238802 [1, 2, 1, 1]  ->  2 /
%e A238802 [1, 2, 3, 1]  ->  3 } ---- 1
%e A238802 [1, 2, 3, 4]  ->  4 } ---- 1
%e A238802 Thus row 4 = [0, 5, 3, 1, 1].
%e A238802 Triangle T(n,k) begins:
%e A238802 00:   1;
%e A238802 01:   0,    1;
%e A238802 02:   0,    1,    1;
%e A238802 03:   0,    2,    1,    1;
%e A238802 04:   0,    5,    3,    1,   1;
%e A238802 05:   0,   13,    8,    3,   1,  1;
%e A238802 06:   0,   38,   24,    9,   3,  1,  1;
%e A238802 07:   0,  116,   74,   28,   9,  3,  1,  1;
%e A238802 08:   0,  382,  246,   93,  29,  9,  3,  1,  1;
%e A238802 09:   0, 1310,  848,  321,  98, 29,  9,  3,  1,  1;
%e A238802 10:   0, 4748, 3088, 1168, 350, 99, 29,  9,  3,  1,  1;
%p A238802 b:= proc(n, l) option remember; `if`(n=0, 1,
%p A238802        b(n-1, [l[], 1]) +add(`if`(i=1 or l[i-1]>l[i],
%p A238802        b(n-1, subsop(i=l[i]+1, l)), 0), i=1..nops(l)))
%p A238802     end:
%p A238802 T:= (n, k)-> `if`(n=k, 1, `if`(k=0, 0, b(n-k-1, [2, 1$(k-1)]))):
%p A238802 seq(seq(T(n, k), k=0..n), n=0..14);
%t A238802 b[n_, l_] := b[n, l] = If[n == 0, 1, b[n-1, Append[l, 1]] + Sum[If[i == 1 || l[[i-1]] > l[[i]], b[n-1, ReplacePart[l, i -> l[[i]]+1]], 0], {i, 1, Length[l]}]]; T[n_, k_] := If[n == k, 1, If[k == 0, 0, b[n-k-1, Join[{2}, Table[1, {k-1}]]]]]; Table[Table[T[n, k], {k, 0, n}], {n, 0, 14}] // Flatten (* _Jean-François Alcover_, Jan 06 2015, translated from Maple *)
%K A238802 nonn,tabl
%O A238802 0,8
%A A238802 _Joerg Arndt_ and _Alois P. Heinz_, Mar 05 2014