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.
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, 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, 4748, 3088, 1168, 350, 99, 29, 9, 3, 1, 1, 0, 17848, 11644, 4404, 1302, 356, 99, 29, 9, 3, 1, 1
Offset: 0
Examples
The 10 tableaux with n=4 cells sorted by the length of the maximal consecutive sequence 1,2,...,k in the first column are: :[1 2] [1 2] [1 2 3] [1 2 4] [1 2 3 4]:[1 3] [1 3] [1 3 4]:[1 4]:[1]: :[3] [3 4] [4] [3] :[2] [2 4] [2] :[2] :[2]: :[4] :[4] :[3] :[3]: : : : :[4]: : -----------------1----------------- : --------2-------- : -3- : 4 : Their corresponding ballot sequences are: [1, 1, 2, 3] -> 1 \ [1, 1, 2, 2] -> 1 \ [1, 1, 1, 2] -> 1 } -- 5 [1, 1, 2, 1] -> 1 / [1, 1, 1, 1] -> 1 / [1, 2, 1, 3] -> 2 \ [1, 2, 1, 2] -> 2 } --- 3 [1, 2, 1, 1] -> 2 / [1, 2, 3, 1] -> 3 } ---- 1 [1, 2, 3, 4] -> 4 } ---- 1 Thus row 4 = [0, 5, 3, 1, 1]. Triangle T(n,k) begins: 00: 1; 01: 0, 1; 02: 0, 1, 1; 03: 0, 2, 1, 1; 04: 0, 5, 3, 1, 1; 05: 0, 13, 8, 3, 1, 1; 06: 0, 38, 24, 9, 3, 1, 1; 07: 0, 116, 74, 28, 9, 3, 1, 1; 08: 0, 382, 246, 93, 29, 9, 3, 1, 1; 09: 0, 1310, 848, 321, 98, 29, 9, 3, 1, 1; 10: 0, 4748, 3088, 1168, 350, 99, 29, 9, 3, 1, 1;
Links
- Joerg Arndt and Alois P. Heinz, Rows n = 0..50, flattened
- Wikipedia, Young tableau
Programs
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Maple
b:= proc(n, l) option remember; `if`(n=0, 1, b(n-1, [l[], 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))) end: T:= (n, k)-> `if`(n=k, 1, `if`(k=0, 0, b(n-k-1, [2, 1$(k-1)]))): seq(seq(T(n, k), k=0..n), n=0..14); -
Mathematica
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 *)
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