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For partitions: A045931, ranked by A325698, strict A239241 (conj A352129).
The version Without multiplicity: A242821, for partitions is A045931, A241638 (ranked by A325698, strict A239241 (conjugate A352129A325700).
Counting only distinct parts gives A242821, for partitions A241638 (ranked by A325700).
These compositions are ranked by A351598A355321.
Cf. A000070, A000700, A000712, A001405, A024619, A026424, A028260, A027193, A035363, `~A066207, `~A066208, A097613, A195017, A240009, A277579 (ranked by A349157).
These compositions are ranked by A351598.
A108950/A108949 count partitions with more odd/even parts, weak A130780/A171966.
Cf. A000070 ptns_altsum_1, A000712 ptns_2kinds, A026424 odd_om, A028260 ev_om, A027193 ptns_odd_len, A035363 ptns_use_ev, A066207 all_ev_prix, A066208 all_odd_prix, A195017 ev_minus_odd_prix.
Cf. A103919 tri_ptns_altsum, A350947 h_allfour_eq, A277579 ptns_num_ev_eq_num_odd_conj.
A130780/A171966 count partitions with more or as many odd/even parts.
Cf. A000070, A000712, A026424, A028260, A027193, A035363, `~A066207, `~A066208, A195017.
From Gus Wiseman, Jun 26 2022: (Start)
The a(0) = 1 through a(7) = 6 compositions (empty columns indicated by dots):
() . . (12) . (14) (1122) (16)
(21) (23) (1212) (25)
(32) (1221) (34)
(41) (2112) (43)
(2121) (52)
(2211) (61)
(End)
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], Count[#, _?EvenQ]==Count[#, _?OddQ]&]], {n, 0, 15}] (* Gus Wiseman, Jun 26 2022 *)
Cf. A045931.
The version for partitions is A045931, ranked by A325698, strict A239241 (conjugate A352129).
Counting only distinct parts gives A242821, for partitions A241638 (ranked by A325700).
A047993 counts balanced partitions, ranked by A106529.
A108950/A108949 count partitions with more odd/even parts, weak A130780/A171966.
Cf. A000070 ptns_altsum_1, A000712 ptns_2kinds, A026424 odd_om, A028260 ev_om, A027193 ptns_odd_len, A035363 ptns_use_ev, A066207 all_ev_prix, A066208 all_odd_prix, A195017 ev_minus_odd_prix.
Cf. A103919 tri_ptns_altsum, A350947 h_allfour_eq, A277579 ptns_num_ev_eq_num_odd_conj.
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a(n) = Sum_{k=floor(n/3)..floor(n/2)} binomialC(2*n-4*k, n-2*k)*binomialC(n-1-k, 2*n-4*k-1).
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Column k=0 of A242498.
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