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Number of integer partitions of n with more adjacent equal parts than distinct parts.
17

%I #9 Feb 21 2023 07:34:04

%S 0,0,0,1,1,1,3,4,7,10,12,18,28,36,52,68,92,119,161,204,269,355,452,

%T 571,738,921,1167,1457,1829,2270,2834,3483,4314,5300,6502,7932,9665,

%U 11735,14263,17227,20807,25042,30137,36099,43264,51646,61608,73291,87146,103296

%N Number of integer partitions of n with more adjacent equal parts than distinct parts.

%C None of these partitions is strict.

%C Also the number of integer partitions of n which, after appending 0, have first differences of median 0.

%e The a(3) = 1 through a(9) = 10 partitions:

%e (111) (1111) (11111) (222) (22111) (2222) (333)

%e (21111) (31111) (22211) (22221)

%e (111111) (211111) (41111) (33111)

%e (1111111) (221111) (51111)

%e (311111) (222111)

%e (2111111) (411111)

%e (11111111) (2211111)

%e (3111111)

%e (21111111)

%e (111111111)

%e For example, the partition y = (4,4,3,1,1,1,1) has 0-appended differences (0,1,2,0,0,0,0), with median 0, so y is counted under a(15).

%t Table[Length[Select[IntegerPartitions[n], Length[#]>2*Length[Union[#]]&]],{n,0,30}]

%Y The non-prepended version is A237363.

%Y These partitions have ranks A360558.

%Y For any integer median (not just 0) we have A360688.

%Y A000041 counts integer partitions, strict A000009.

%Y A008284 counts partitions by number of parts.

%Y A116608 counts partitions by number of distinct parts.

%Y A325347 counts partitions w/ integer median, strict A359907, ranks A359908.

%Y A359893 and A359901 count partitions by median, odd-length A359902.

%Y Cf. A000975, A027193, A067538, A102627, A240219, A359894, A360071, A360244, A360555, A360556.

%K nonn

%O 0,7

%A _Gus Wiseman_, Feb 20 2023