cp's OEIS Frontend

A362609 Number of integer partitions of n with more than one part of least multiplicity.

%I A362609 #7 May 02 2023 16:08:00
%S A362609 0,0,0,1,1,2,4,5,9,14,19,26,42,51,74,103,136,174,246,303,411,523,674,
%T A362609 844,1114,1364,1748,2174,2738,3354,4247,5139,6413,7813,9613,11630,
%U A362609 14328,17169,20958,25180,30497,36401,44025,52285,62834,74626,89111,105374,125662
%N A362609 Number of integer partitions of n with more than one part of least multiplicity.
%C A362609 These are partitions where no part appears fewer times than all of the others.
%e A362609 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 A362609 The a(3) = 1 through a(9) = 14 partitions:
%e A362609   (21)  (31)  (32)  (42)    (43)    (53)     (54)
%e A362609               (41)  (51)    (52)    (62)     (63)
%e A362609                     (321)   (61)    (71)     (72)
%e A362609                     (2211)  (421)   (431)    (81)
%e A362609                             (3211)  (521)    (432)
%e A362609                                     (3221)   (531)
%e A362609                                     (3311)   (621)
%e A362609                                     (4211)   (3321)
%e A362609                                     (32111)  (4221)
%e A362609                                              (4311)
%e A362609                                              (5211)
%e A362609                                              (42111)
%e A362609                                              (222111)
%e A362609                                              (321111)
%t A362609 Table[Length[Select[IntegerPartitions[n],Count[Length/@Split[#],Min@@Length/@Split[#]]>1&]],{n,0,30}]
%Y A362609 For parts instead of multiplicities we have A117989, ranks A283050.
%Y A362609 For median instead of co-mode we have A238479, complement A238478.
%Y A362609 These partitions have ranks A362606.
%Y A362609 For mode instead of co-mode we have A362607, ranks A362605.
%Y A362609 For mode complement instead of co-mode we have A362608, ranks A356862.
%Y A362609 The complement is counted by A362610, ranks A359178.
%Y A362609 A000041 counts integer partitions.
%Y A362609 A275870 counts collapsible partitions.
%Y A362609 A359893 counts partitions by median.
%Y A362609 A362611 counts modes in prime factorization, co-modes A362613.
%Y A362609 A362614 counts partitions by number of modes, co-modes A362615.
%Y A362609 Cf. A002865, A008284, A053263, A098859, A304442, A353864, A360071, A362612.
%K A362609 nonn
%O A362609 0,6
%A A362609 _Gus Wiseman_, Apr 30 2023