A337583 Irregular triangle read by rows: T(n, k) is the number of integer multisets (partitions) that match the multiplicity multiset of exactly k partitions of n.
1, 1, 2, 3, 5, 3, 2, 9, 1, 7, 1, 2, 12, 2, 2, 12, 2, 2, 2, 15, 3, 3, 3, 15, 5, 3, 0, 3, 0, 1, 26, 8, 2, 1, 2, 0, 1, 1, 23, 7, 2, 4, 1, 3, 0, 1, 0, 0, 1, 28, 9, 4, 5, 2, 2, 2, 0, 0, 1, 1, 33, 11, 3, 4, 2, 3, 3, 1, 0, 1, 0, 1, 0, 0, 0, 0, 1, 45, 10, 8, 4, 4, 1, 4, 1, 1, 0, 1, 0, 1, 1, 0, 0, 1, 40, 18, 7, 3, 5
Offset: 0
Examples
T(5, 1) = 3, T(5, 2) = 2: The partitions of 5 present A088887(5) = 5 different multiplicity multisets. Three of them are attained by a single partition of 5 (for instance, (3, 1) comes from (2, 1, 1, 1) only), whereas (1, 1) and (2, 1) arise from two partitions of 5 each (namely, (4, 1) and (3, 2) for the first and (3, 1, 1) and (2, 2, 1) for the second).
Triangle begins:
k: 1 2 3 4
-------
n=0: 1
n=1: 1
n=2: 2
n=3: 3
n=4: 5
n=5: 3 2
n=6: 9 1
n=7: 7 1 2
n=8: 12 2 2
n=9: 12 2 2 2
Links
- Álvar Ibeas, Rows until n=66, flattened
- Álvar Ibeas, Rows until n=19
Formula
Sum_{k >= 1} k * T(n, k) = A000041(n).