cp's OEIS Frontend

A364193 Number of integer partitions of n where the least part is the unique mode.

%I A364193 #7 Jul 17 2023 17:59:34
%S A364193 0,1,2,2,4,4,7,9,13,17,24,32,43,58,75,97,130,167,212,274,346,438,556,
%T A364193 695,865,1082,1342,1655,2041,2511,3067,3756,4568,5548,6728,8130,9799,
%U A364193 11810,14170,16980,20305,24251,28876,34366,40781,48342,57206,67597,79703
%N A364193 Number of integer partitions of n where the least part is the unique mode.
%C A364193 A mode in a multiset is an element that appears at least as many times as each of the others. For example, the modes in {a,a,b,b,b,c,d,d,d} are {b,d}.
%e A364193 The a(1) = 1 through a(8) = 13 partitions:
%e A364193   (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
%e A364193        (11)  (111)  (22)    (311)    (33)      (322)      (44)
%e A364193                     (211)   (2111)   (222)     (511)      (422)
%e A364193                     (1111)  (11111)  (411)     (3211)     (611)
%e A364193                                      (3111)    (4111)     (2222)
%e A364193                                      (21111)   (22111)    (4211)
%e A364193                                      (111111)  (31111)    (5111)
%e A364193                                                (211111)   (32111)
%e A364193                                                (1111111)  (41111)
%e A364193                                                           (221111)
%e A364193                                                           (311111)
%e A364193                                                           (2111111)
%e A364193                                                           (11111111)
%t A364193 Table[If[n==0,0,Length[Select[IntegerPartitions[n], Last[Length/@Split[#]]>Max@@Most[Length/@Split[#]]&]]],{n,0,30}]
%Y A364193 For greatest part and multiple modes we have A171979.
%Y A364193 Allowing multiple modes gives A240303.
%Y A364193 For greatest instead of least part we have A362612, ranks A362616.
%Y A364193 For mean instead of least part we have A363723.
%Y A364193 These partitions have ranks A364160.
%Y A364193 A000041 counts integer partitions.
%Y A364193 A362611 counts modes in prime factorization, A362613 co-modes.
%Y A364193 A362614 counts partitions by number of modes, co-modes A362615.
%Y A364193 A363486 gives least mode in prime indices, A363487 greatest.
%Y A364193 A363952 counts partitions by low mode, A363953 high.
%Y A364193 Ranking and counting partitions:
%Y A364193 - A356862 = unique mode, counted by A362608
%Y A364193 - A359178 = unique co-mode, counted by A362610
%Y A364193 - A362605 = multiple modes, counted by A362607
%Y A364193 - A362606 = multiple co-modes, counted by A362609
%Y A364193 Cf. A002865, A008284, A070003, A098859, A102750, A237984, A327472, A360015.
%K A364193 nonn
%O A364193 0,3
%A A364193 _Gus Wiseman_, Jul 16 2023