A210950 Triangle read by rows: T(n,k) = number of parts in the k-th column of the partitions of n but with the partitions aligned to the right margin.

1, 1, 2, 1, 2, 3, 1, 2, 4, 5, 1, 2, 4, 6, 7, 1, 2, 4, 7, 10, 11, 1, 2, 4, 7, 11, 14, 15, 1, 2, 4, 7, 12, 17, 21, 22, 1, 2, 4, 7, 12, 18, 25, 29, 30, 1, 2, 4, 7, 12, 19, 28, 36, 41, 42, 1, 2, 4, 7, 12, 19, 29, 40, 50, 55, 56, 1, 2, 4, 7, 12, 19, 30, 43

Offset

1,3

Comments

Index of the first partition of n that has k parts, when the partitions of n are listed in reverse lexicographic order, as in Mathematica's IntegerPartitions[n]. - Clark Kimberling, Oct 16 2023

Links

Table of n, a(n) for n=1..74.

Formula

T(n,k) = Sum_{j=1..n} A210951(j,k).

Example

For n = 6 the partitions of 6 aligned to the right margin look like this:
.
.                      6
.                  3 + 3
.                  4 + 2
.              2 + 2 + 2
.                  5 + 1
.              3 + 2 + 1
.              4 + 1 + 1
.          2 + 2 + 1 + 1
.          3 + 1 + 1 + 1
.      2 + 1 + 1 + 1 + 1
.  1 + 1 + 1 + 1 + 1 + 1
.
The number of parts in columns 1-6 are
.  1,  2,  4,  7, 10, 11, the same as the 6th row of triangle.
Triangle begins:
  1;
  1, 2;
  1, 2, 3;
  1, 2, 4, 5;
  1, 2, 4, 6, 7;
  1, 2, 4, 7, 10, 11;
  1, 2, 4, 7, 11, 14, 15;
  1, 2, 4, 7, 12, 17, 21, 22;
  1, 2, 4, 7, 12, 18, 25, 29, 30;
  1, 2, 4, 7, 12, 19, 28, 36, 41, 42;
  1, 2, 4, 7, 12, 19, 29, 40, 50, 55, 56;
  1, 2, 4, 7, 12, 19, 30, 43, 58, 70, 76, 77;

Mathematica

m[n_, k_] := Length[IntegerPartitions[n][[k]]]; c[n_] := PartitionsP[n];
t[n_, h_] := Select[Range[c[n]], m[n, #] == h &, 1];
Column[Table[t[n, h], {n, 1, 20}, {h, 1, n}]]
 (* Clark Kimberling, Oct 16 2023 *)

Crossrefs

Mirror of A058399. Row sums give A006128. Right border gives A000041, n >= 1. Rows converge to A000070.

Cf. A135010, A194714, A210945, A210951, A210952, A210953, A210970.

Sequence in context: A116925 A309010 A308500 * A214314 A209435 A263744

Adjacent sequences: A210947 A210948 A210949 * A210951 A210952 A210953

Keyword

nonn,tabl

Author

Omar E. Pol, Apr 22 2012

Status

approved