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%I #14 May 04 2023 14:57:32
%S 0,1,2,2,4,5,7,11,16,21,29,43,54,78,102,131,175,233,295,389,490,623,
%T 794,1009,1255,1579,1967,2443,3016,3737,4569,5627,6861,8371,10171,
%U 12350,14901,18025,21682,26068,31225,37415,44617,53258,63313,75235,89173,105645
%N Number of integer partitions of n having a 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 Andrew Howroyd, <a href="/A362608/b362608.txt">Table of n, a(n) for n = 0..1000</a>
%F G.f.: Sum_{m>=1} (Sum_{j>=1} x^(j*m)*(1 - x^j)/(1 - x^(j*m))) * (Product_{j>=1} (1 - x^(j*m))/(1 - x^j)). - _Andrew Howroyd_, May 04 2023
%e The partition (3,3,2,1) has greatest multiplicity 2, and a unique part of multiplicity 2 (namely 3), so is counted under a(9).
%e The a(1) = 1 through a(7) = 11 partitions:
%e (1) (2) (3) (4) (5) (6) (7)
%e (11) (111) (22) (221) (33) (322)
%e (211) (311) (222) (331)
%e (1111) (2111) (411) (511)
%e (11111) (3111) (2221)
%e (21111) (3211)
%e (111111) (4111)
%e (22111)
%e (31111)
%e (211111)
%e (1111111)
%t Table[Length[Select[IntegerPartitions[n],Length[Commonest[#]]==1&]],{n,0,30}]
%o (PARI) seq(n) = my(A=O(x*x^n)); Vec(sum(m=1, n, sum(j=1, n\m, x^(j*m)*(1-x^j)/(1 - x^(j*m)), A)*prod(j=1, n\m, (1 - x^(j*m))/(1 - x^j) + A)/prod(j=n\m+1, n, 1 - x^j + A)), -(n+1)) \\ _Andrew Howroyd_, May 04 2023
%Y For parts instead of multiplicities we have A000041(n-1), ranks A102750.
%Y For median instead of mode we have A238478, complement A238479.
%Y These partitions have ranks A356862.
%Y The complement is counted by A362607, ranks A362605.
%Y For co-mode complement we have A362609, ranks A362606.
%Y For co-mode we have A362610, ranks A359178.
%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, A237984, A304442, A327472, A360071, A360687, A362612.
%K nonn
%O 0,3
%A _Gus Wiseman_, Apr 30 2023
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