A391225 - OEIS

%I #8 Dec 08 2025 10:44:00

%S 1,2,2,4,3,8,6,14,13,25,24,45,45,74,82,125,139,204,233,329,382,517,

%T 609,811,957,1242,1478,1892,2249,2839,3381,4221,5024,6205,7386,9055,

%U 10751,13078,15513,18760,22198,26687,31521,37710,44440,52913,62228,73804,86590

%N Number of integer partitions of n such that the least and greatest parts are both even or both odd.

%e The a(1) = 1 through a(9) = 13 partitions:

%e   (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)         (9)

%e        (11)  (111)  (22)    (311)    (33)      (331)      (44)        (333)

%e                     (31)    (11111)  (42)      (511)      (53)        (432)

%e                     (1111)           (51)      (3211)     (62)        (531)

%e                                      (222)     (31111)    (71)        (711)

%e                                      (321)     (1111111)  (422)       (3321)

%e                                      (3111)               (521)       (5211)

%e                                      (111111)             (2222)      (32211)

%e                                                           (3221)      (33111)

%e                                                           (3311)      (51111)

%e                                                           (5111)      (321111)

%e                                                           (32111)     (3111111)

%e                                                           (311111)    (111111111)

%e                                                           (11111111)

%t Table[Length[Select[IntegerPartitions[n],EvenQ[First[#]+Last[#]]&]],{n,30}]

%Y For even least part we have A026805, ranks A340933, strict A026833.

%Y For even length we have A027187, ranks A028260, strict A067661.

%Y For even greatest part we have A027187, ranks A244990, strict A067661.

%Y For even sum of prime indices we have A300061, counted by A058696.

%Y For both parts even we have A325346, ranks A391232, strict A391233.

%Y The complement is counted by A390092, ranks A390988.

%Y These partitions are ranked by A391226.

%Y The strict case is A391227, complement counted by A390746.

%Y For product instead of sum we have A391229, counted by A391230, strict A391231.

%Y A026424 ranks partitions of odd length, counted by A027193, strict A067659.

%Y A244991 ranks partitions of odd maximum, counted by A027193, strict A026837.

%Y A390093 ranks partitions of odd min and max, counted by A325338, strict A391228.

%Y A340932 ranks partitions of odd minimum, counted by A026804, strict A026832.

%Y A066207 lists products of even-indexed primes, counted by A035363, strict A035457.

%Y A066208 lists products of odd-indexed primes, counted by A000009, strict A000700.

%Y A390430 gives least prime index plus greatest prime index, for differences A243055.

%Y Cf. A005843, A101707, A160786, A257992, A300272, A340832.

%K nonn

%O 1,2

%A _Gus Wiseman_, Dec 07 2025