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%I #7 May 02 2023 16:08:00
%S 0,0,0,1,1,2,4,5,9,14,19,26,42,51,74,103,136,174,246,303,411,523,674,
%T 844,1114,1364,1748,2174,2738,3354,4247,5139,6413,7813,9613,11630,
%U 14328,17169,20958,25180,30497,36401,44025,52285,62834,74626,89111,105374,125662
%N Number of integer partitions of n with more than one part of least multiplicity.
%C These are partitions where no part appears fewer times than all of the others.
%e The partition (4,2,2,1) has least multiplicity 1, and two parts of multiplicity 1 (namely 1 and 4), so is counted under a(9).
%e The a(3) = 1 through a(9) = 14 partitions:
%e (21) (31) (32) (42) (43) (53) (54)
%e (41) (51) (52) (62) (63)
%e (321) (61) (71) (72)
%e (2211) (421) (431) (81)
%e (3211) (521) (432)
%e (3221) (531)
%e (3311) (621)
%e (4211) (3321)
%e (32111) (4221)
%e (4311)
%e (5211)
%e (42111)
%e (222111)
%e (321111)
%t Table[Length[Select[IntegerPartitions[n],Count[Length/@Split[#],Min@@Length/@Split[#]]>1&]],{n,0,30}]
%Y For parts instead of multiplicities we have A117989, ranks A283050.
%Y For median instead of co-mode we have A238479, complement A238478.
%Y These partitions have ranks A362606.
%Y For mode instead of co-mode we have A362607, ranks A362605.
%Y For mode complement instead of co-mode we have A362608, ranks A356862.
%Y The complement is counted by A362610, ranks A359178.
%Y A000041 counts integer partitions.
%Y A275870 counts collapsible partitions.
%Y A359893 counts partitions by median.
%Y A362611 counts modes in prime factorization, co-modes A362613.
%Y A362614 counts partitions by number of modes, co-modes A362615.
%Y Cf. A002865, A008284, A053263, A098859, A304442, A353864, A360071, A362612.
%K nonn
%O 0,6
%A _Gus Wiseman_, Apr 30 2023
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