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A362612
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Number of integer partitions of n such that the greatest part is the unique mode.
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35
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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, 101, 115, 135, 151, 176, 200, 228, 261, 300, 336, 385, 439, 498, 561, 641, 717, 818, 921, 1036, 1166, 1321, 1477, 1667, 1867, 2099, 2346, 2640, 2944, 3303, 3684
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OFFSET
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0,3
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COMMENTS
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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}.
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LINKS
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FORMULA
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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
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EXAMPLE
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The a(1) = 1 through a(10) = 7 partitions (A = 10):
1 2 3 4 5 6 7 8 9 A
11 111 22 221 33 331 44 333 55
1111 11111 222 2221 332 441 442
111111 1111111 2222 3321 3331
22211 22221 22222
11111111 111111111 222211
1111111111
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MATHEMATICA
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Table[Length[Select[IntegerPartitions[n], Commonest[#]=={Max[#]}&]], {n, 0, 30}]
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PROG
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(PARI)
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))}
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CROSSREFS
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These partitions have ranks A362616.
A275870 counts collapsible partitions.
A359893 counts partitions by median.
A362611 counts modes in prime factorization.
Cf. A002865, A008284, A070003, A098859, A102750, A237984, A238478, A238479, A327472, A362609, A362610.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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