%I #13 Apr 03 2024 21:21:53

%S 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 101,115,135,151,176,200,228,261,300,336,385,439,498,561,641,717,818,

%U 921,1036,1166,1321,1477,1667,1867,2099,2346,2640,2944,3303,3684

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

%C 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 John Tyler Rascoe, <a href="/A362612/b362612.txt">Table of n, a(n) for n = 0..500</a>

%F 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 The a(1) = 1 through a(10) = 7 partitions (A = 10):

%e 1 2 3 4 5 6 7 8 9 A

%e 11 111 22 221 33 331 44 333 55

%e 1111 11111 222 2221 332 441 442

%e 111111 1111111 2222 3321 3331

%e 22211 22221 22222

%e 11111111 111111111 222211

%e 1111111111

%t Table[Length[Select[IntegerPartitions[n],Commonest[#]=={Max[#]}&]],{n,0,30}]

%o (PARI)

%o 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 A_x(60) \\ _John Tyler Rascoe_, Apr 03 2024

%Y For median instead of mode we have A053263, complement A237821.

%Y These partitions have ranks A362616.

%Y A000041 counts integer partitions.

%Y A275870 counts collapsible partitions.

%Y A359893 counts partitions by median.

%Y A362607 counts partitions with more than one mode, ranks A362605.

%Y A362608 counts partitions with a unique mode, ranks A356862.

%Y A362611 counts modes in prime factorization.

%Y A362614 counts partitions by number of modes, co-modes A362615.

%Y Cf. A002865, A008284, A070003, A098859, A102750, A237984, A238478, A238479, A327472, A362609, A362610.

%K nonn

%O 0,3

%A _Gus Wiseman_, May 03 2023