A180681 T(n,k) is the sum of the hook lengths over the partitions of n with exactly k parts.
1, 3, 3, 6, 5, 6, 10, 16, 8, 10, 15, 23, 22, 12, 15, 21, 47, 44, 30, 17, 21, 28, 62, 74, 56, 40, 23, 28, 36, 104, 115, 114, 71, 52, 30, 36, 45, 130, 196, 162, 139, 89, 66, 38, 45, 55, 195, 268, 286, 227, 169, 110, 82, 47, 55, 66, 235, 395, 407, 369, 269, 204, 134, 100, 57, 66
Offset: 1
Examples
T(5,3) = 22 since the partitions of 5 in 3 parts are 221 and 311, with hook lengths {{2,4}, {1,3}, {1}} and {{1,2,5}, {2}, {1}} summing to 22.
Triangle T(n,k) begins:
1;
3, 3;
6, 5, 6;
10, 16, 8, 10;
15, 23, 22, 12, 15;
21, 47, 44, 30, 17, 21;
28, 62, 74, 56, 40, 23, 28;
36, 104, 115, 114, 71, 52, 30, 36;
45, 130, 196, 162, 139, 89, 66, 38, 45;
55, 195, 268, 286, 227, 169, 110, 82, 47, 55;
Links
- Alois P. Heinz, Rows n = 1..200, flattened
Programs
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Maple
f:= n-> (n-1)*n/2: b:= proc(n, i) option remember; `if`(n=0 or i=1, [1, n+f(n)], b(n, i-1)+(p-> p+[0, p[1]*(n+f(i))])(b(n-i, min(n-i, i)))) end: T:= (n, k)-> (p-> p[1]*(n+f(k))+p[2])(b(n-k, min(n-k, k))): seq(seq(T(n, k), k=1..n), n=1..14); # Alois P. Heinz, Mar 20 2018 -
Mathematica
(*Needs["DiscreteMath`Combinatorica`"]; hooklength[(p_)?PartitionQ] := Block[{ferr = (PadLeft[1 + 0*Range[ #1], Max[p]] & ) /@ p}, DeleteCases[(Rest[FoldList[Plus, 0, #1]] & ) /@ ferr + Reverse /@ Reverse[Transpose[(Rest[FoldList[Plus, 0, #1]] & ) /@ Reverse[Reverse /@ Transpose[ferr]]]], 0, {2}] - 1]; partitionexact[n_, m_] := TransposePartition /@ (Prepend[ #1, m] & ) /@ Partitions[n - m, m] *); Table[Tr[ Tr[ Flatten[hooklength[ # ]]] &/@ partitionexact[n,k] ] ,{n,16},{k,n}] (* Second program: *) Table[p = IntegerPartitions[n, {k}]; Total@Table[y = Table[Boole[p[[l]][[i]] >= j], {i, k}, {j, n}]; Total[Table[Total[{y[[i, j ;; n]], y[[i + 1 ;; k, j]]}, 2], {i, k}, {j, n}], 2], {l, Length[p]}], {n, 11}, {k, n}] // Flatten (* Robert Price, Jun 19 2020 *) f[n_] := n(n-1)/2; b[n_, i_] := b[n, i] = If[n == 0 || i == 1, {1, n + f[n]}, b[n, i - 1] + Function[p, p + {0, p[[1]] (n + f[i])}][b[n - i, Min[n - i, i]]]]; T[n_, k_] := Function[p, p[[1]] (n + f[k]) + p[[2]]][b[n-k, Min[n-k, k]]]; Table[Table[T[n, k], {k, 1, n}], {n, 1, 14}] // Flatten (* Jean-François Alcover, Dec 12 2020, after Alois P. Heinz *)
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