A337584 -id:A337584 - cp's OEIS frontend

A337586 Triangle read by rows: T(n, k) is the number of integer multisets of size k (partitions of k) for which the number of partitions of n with matching multiplicity multiset is odd (n >= 1, 1 <= k <= n).

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 0, 0, 1, 1, 1, 1, 3, 2, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 3, 3, 2, 1, 1, 1, 0, 3, 0, 3, 3, 2, 1, 1, 1, 1, 0, 3, 3, 3, 3, 2, 1, 1, 1, 1, 2, 3, 2, 1, 5, 3, 2, 1, 1, 1, 2, 2, 1, 3, 7, 3, 5, 3, 2, 1, 1, 1, 0, 0, 2, 2, 2, 5, 3, 5, 3, 2, 1, 1, 1, 1, 0, 3, 3, 4, 5, 5, 3, 5, 3

Offset: 1

Álvar Ibeas, Sep 02 2020

The relevant partitions of n have exactly k parts.

The number of multiplicity multisets of size k met by a positive even number of partitions of n is A337584(n, k) - T(n, k).

			The 3 = A008284(6, 2) partitions of 6 into 2 parts show 2 = A337584(6, 2) different multiplicity multisets: (1, 1) is attained by two of those partitions ((5, 1) and (4, 2)) and the other (2) just by one, (3, 3). Then, T(6, 2) = 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 0 0 1 1
n=6:  1 1 3 2 1 1
n=7:  1 1 2 1 2 1 1
n=8:  1 2 1 3 3 2 1 1
n=9:  1 0 3 0 3 3 2 1 1
n=10: 1 1 0 3 3 3 3 2 1 1

Álvar Ibeas, First 72 rows, flattened
Álvar Ibeas, First 30 rows

Cf. A008284, A337585 (row sums), A337584.

T(n, k) == A008284(n, k) (mod 2).

If k > (2*n+1)/3, T(n, k) = A337585(n - k).

A344680 Number of partitions of n that are also multiplicity multiset of a partition of 2n.

Original entry on oeis.org

1, 1, 2, 3, 4, 4, 8, 8, 12, 14, 18, 21, 30, 33, 41, 49, 62, 70, 86, 98, 116, 133, 160, 181, 214, 237, 282, 311, 364, 407, 466, 522, 600, 652, 761, 815, 937, 1038, 1179, 1271, 1442, 1577, 1762, 1930, 2158, 2311, 2636, 2831, 3146, 3402, 3784, 4057, 4537, 4869, 5365, 5745, 6370, 6802, 7562, 8061, 8785, 9471, 10410

Offset: 0

Alois P. Heinz, Aug 17 2021

nonn

			a(0) = 1: [].
a(1) = 1: [1].
a(2) = 2: [2], [1,1].
a(3) = 3: [3], [1,2], [1,1,1].
a(4) = 4: [4], [1,3], [2,2], [1,1,2].
a(5) = 4: [5], [1,4], [1,1,3], [1,2,2].
a(6) = 8: [6], [1,5], [2,4], [3,3], [1,1,4], [1,2,3], [2,2,2], [1,1,1,3].
a(7) = 8: [7], [1,6], [1,1,5], [1,2,4], [1,3,3], [2,2,3], [1,1,1,4], [1,1,2,3].

Cf. A088887, A337584.

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:
a:= n-> nops(select(l-> add(i, i=l)=n, b(2*n$2))):
seq(a(n), n=0..30);

a(n) = A337584(2n,n).

cp's OEIS Frontend

A337586 Triangle read by rows: T(n, k) is the number of integer multisets of size k (partitions of k) for which the number of partitions of n with matching multiplicity multiset is odd (n >= 1, 1 <= k <= n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula