OFFSET
0,10
COMMENTS
T(0,0) = 1 by convention.
Also the number of ballot sequences of length n where k is the position of the last occurrence of the minimal value.
Diagonal: T(n,n) = A000085(n-1) for n>=1.
Columns k=0-10 give: A000007, A000012 for n>0, A001405(n-2) for n>1, A245001, A245002, A245003, A245004, A245005, A245006, A245007, A245008.
T(2n,n) gives A245000.
Row sums give A000085.
LINKS
Joerg Arndt and Alois P. Heinz, Rows n = 0..50, flattened
Wikipedia, Young tableau
EXAMPLE
The 10 tableaux with n=4 cells sorted by last value in the first row:
:[1]:[1 2] [1 2]:[1 3] [1 3] [1 2 3]:[1 4] [1 2 4] [1 3 4] [1 2 3 4]:
:[2]:[3] [3 4]:[2] [2 4] [4] :[2] [3] [2] :
:[3]:[4] :[4] :[3] :
:[4]: : : :
: 1 : ----2---- : --------3-------- : --------------4-------------- :
Their corresponding ballot sequences are: [1,2,3,4], [1,1,2,3], [1,1,2,2], [1,2,1,3], [1,2,1,2], [1,1,1,2], [1,2,3,1], [1,1,2,1], [1,2,1,1], [1,1,1,1]. Thus row 4 = [0, 1, 2, 3, 4].
Triangle T(n,k) begins:
00: 1;
01: 0, 1;
02: 0, 1, 1;
03: 0, 1, 1, 2;
04: 0, 1, 2, 3, 4;
05: 0, 1, 3, 5, 7, 10;
06: 0, 1, 6, 10, 14, 19, 26;
07: 0, 1, 10, 19, 29, 41, 56, 76;
08: 0, 1, 20, 41, 66, 96, 132, 176, 232;
09: 0, 1, 35, 86, 152, 232, 327, 441, 582, 764;
10: 0, 1, 70, 197, 374, 596, 863, 1181, 1563, 2031, 2620;
MAPLE
b:= proc(n, l) option remember; `if`(n=0, 1, add(`if`(
i=1 or l[i-1]>l[i], b(n-1, subsop(i=l[i]+1, l)), 0),
i=1..nops(l)) +(p-> p+(x^(1+add(j, j=l))-1)*
coeff(p, x, 0))(b(n-1, [l[], 1])))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n, [])):
seq(T(n), n=0..12);
MATHEMATICA
b[n_, l_List] := b[n, l] = If[n == 0, 1, Sum[If[i == 1 || l[[i-1]] > l[[i]], b[n-1, ReplacePart[l, i -> l[[i]]+1]], 0], {i, 1, Length[l]}] + Function[{p}, p + (x^(1+Total[l])-1)*Coefficient[p, x, 0]][b[n-1, Append[l, 1]]]]; T[n_] := Function[{p}, Table[Coefficient[p, x, i], {i, 0, n}]][b[n, {}]]; Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Jan 13 2015, translated from Maple *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Joerg Arndt and Alois P. Heinz, Mar 05 2014
STATUS
approved