A337584 Triangle read by rows: T(n, k) is the number of integer multisets of size k (partitions of k) that match the multiplicity multiset of some partition of n (n >= 1, 1 <= k <= n).
1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 2, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 2, 2, 4, 3, 2, 1, 1, 1, 1, 3, 2, 4, 3, 2, 1, 1, 1, 2, 2, 4, 4, 4, 3, 2, 1, 1, 1, 1, 2, 3, 5, 3, 5, 3, 2, 1, 1, 1, 2, 3, 5, 5, 8, 5, 5, 3, 2, 1, 1, 1, 1, 2, 3, 5, 5, 8, 5, 5, 3, 2, 1, 1, 1, 2, 2, 4, 5, 7, 8, 8, 5, 5, 3
Offset: 1
Examples
There is no partition of 5 with multiplicity multiset (3) or (1, 1, 1).
Indeed, both (2 = A008284(5, 3)) partitions of 5 into 3 parts (namely, (3, 1, 1) and (2, 2, 1)) have multiplicities (2, 1). Therefore, T(5, 3) = 1.
Triangle begins:
k: 1 2 3 4 5 6 7 8 9 10
--------------------
n=1: 1
n=2: 1 1
n=3: 1 1 1
n=4: 1 2 1 1
n=5: 1 1 1 1 1
n=6: 1 2 3 2 1 1
n=7: 1 1 2 2 2 1 1
n=8: 1 2 2 4 3 2 1 1
n=9: 1 1 3 2 4 3 2 1 1
n=10: 1 2 2 4 4 4 3 2 1 1
Links
- Alois P. Heinz, Rows n = 1..125 (first 71 rows from Álvar Ibeas)
- Álvar Ibeas, First 30 rows
Programs
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Maple
b:= proc(n,i) option remember; `if`(n=0 or i=1, `if`(n=0, {[]}, {[n]}), {b(n, i-1)[], seq(map(x-> sort([x[], j]), b(n-i*j, i-1))[], j=1..n/i)}) end: T:= n-> (p-> seq(coeff(p, x, i), i=1..n))(add(x^add(i, i=t), t=b(n$2))): seq(T(n), n=1..20); # Alois P. Heinz, Aug 17 2021
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