A337583 -id:A337583 - cp's OEIS frontend

A337585 a(n) is the number of integer multisets (partitions) for which the number of partitions of n with matching multiplicity multiset is odd.

1, 1, 2, 3, 5, 3, 9, 9, 14, 14, 18, 22, 31, 27, 37, 42, 65, 61, 83, 82, 111, 110, 142, 147, 187, 190, 230, 242, 296, 319, 358, 412, 471, 505, 600, 595, 753, 781, 895, 921, 1082, 1143, 1272, 1405, 1587, 1632, 1872, 2000, 2263, 2419, 2648, 2799, 3223, 3319, 3723

Offset: 0

Álvar Ibeas, Sep 02 2020

nonn

The number of multiplicity multisets met by a positive even number of partitions of n is A088887(n) - a(n).

The parity of a(n) is A040051(n).

			The partitions of 7 exhibit 10 = A088887(7) different multisets of multiplicities. Except for (3, 1) (met by partitions (4, 1, 1, 1) and (2, 2, 2, 1)), an odd number of partitions of 7 lead to each of them, so a(7) = 9.

Cf. A088887.

a(n) = Sum_{k odd} A337583(n, k).

A337587 a(n) is the maximum size among sets of partitions of n with the same multiplicity multiset.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 4, 7, 8, 11, 11, 17, 17, 23, 23, 30, 36, 44, 56, 65, 79, 91, 110, 124, 146, 165, 189, 212, 245, 280, 343, 393, 465, 530, 623, 698, 815, 910, 1043, 1163, 1325, 1464, 1690, 1898, 2190, 2590, 2981, 3458, 3965, 4568, 5156, 5931, 6689, 7571

Offset: 0

Álvar Ibeas, Sep 02 2020

nonn

			a(7) = 3, cardinality reached by two sets. On the one hand, the multiplicities of the three partitions (6, 1), (5, 2), and (4, 3) are (1, 1). On the other hand, those of (5, 1, 1), (3, 3, 1) and (3, 2, 2) are (2, 1).

Cf. A337583.

cp's OEIS Frontend

A337585 a(n) is the number of integer multisets (partitions) for which the number of partitions of n with matching multiplicity multiset is odd.

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