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A362612 Number of integer partitions of n such that the greatest part is the unique mode.

%I A362612 #13 Apr 03 2024 21:21:53
%S A362612 0,1,2,2,3,3,4,4,6,6,7,9,10,12,15,16,19,23,26,32,37,41,48,58,65,75,88,
%T A362612 101,115,135,151,176,200,228,261,300,336,385,439,498,561,641,717,818,
%U A362612 921,1036,1166,1321,1477,1667,1867,2099,2346,2640,2944,3303,3684
%N A362612 Number of integer partitions of n such that the greatest part is the unique mode.
%C A362612 A mode in a multiset is an element that appears at least as many times as each of the others. For example, the modes of {a,a,b,b,b,c,d,d,d} are {b,d}.
%H A362612 John Tyler Rascoe, <a href="/A362612/b362612.txt">Table of n, a(n) for n = 0..500</a>
%F A362612 G.f.: Sum_{i, j>0} x^(i*j) * Product_{k=1,i-1} ((1-x^(j*k))/(1-x^k)). - _John Tyler Rascoe_, Apr 03 2024
%e A362612 The a(1) = 1 through a(10) = 7 partitions (A = 10):
%e A362612   1  2   3    4     5      6       7        8         9          A
%e A362612      11  111  22    221    33      331      44        333        55
%e A362612               1111  11111  222     2221     332       441        442
%e A362612                            111111  1111111  2222      3321       3331
%e A362612                                             22211     22221      22222
%e A362612                                             11111111  111111111  222211
%e A362612                                                                  1111111111
%t A362612 Table[Length[Select[IntegerPartitions[n],Commonest[#]=={Max[#]}&]],{n,0,30}]
%o A362612 (PARI)
%o A362612 A_x(N)={my(x='x+O('x^N), g=sum(i=1, N, sum(j=1, N/i, x^(i*j)*prod(k=1,i-1,(1-x^(j*k))/(1-x^k))))); concat([0],Vec(g))}
%o A362612 A_x(60) \\ _John Tyler Rascoe_, Apr 03 2024
%Y A362612 For median instead of mode we have A053263, complement A237821.
%Y A362612 These partitions have ranks A362616.
%Y A362612 A000041 counts integer partitions.
%Y A362612 A275870 counts collapsible partitions.
%Y A362612 A359893 counts partitions by median.
%Y A362612 A362607 counts partitions with more than one mode, ranks A362605.
%Y A362612 A362608 counts partitions with a unique mode, ranks A356862.
%Y A362612 A362611 counts modes in prime factorization.
%Y A362612 A362614 counts partitions by number of modes, co-modes A362615.
%Y A362612 Cf. A002865, A008284, A070003, A098859, A102750, A237984, A238478, A238479, A327472, A362609, A362610.
%K A362612 nonn
%O A362612 0,3
%A A362612 _Gus Wiseman_, May 03 2023