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A362612
Number of integer partitions of n such that the greatest part is the unique mode.
35
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
OFFSET
0,3
COMMENTS
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}.
LINKS
FORMULA
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
EXAMPLE
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
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], Commonest[#]=={Max[#]}&]], {n, 0, 30}]
PROG
(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))}
A_x(60) \\ John Tyler Rascoe, Apr 03 2024
CROSSREFS
For median instead of mode we have A053263, complement A237821.
These partitions have ranks A362616.
A000041 counts integer partitions.
A275870 counts collapsible partitions.
A359893 counts partitions by median.
A362607 counts partitions with more than one mode, ranks A362605.
A362608 counts partitions with a unique mode, ranks A356862.
A362611 counts modes in prime factorization.
A362614 counts partitions by number of modes, co-modes A362615.
Sequence in context: A356737 A058747 A034322 * A319169 A050365 A029026
KEYWORD
nonn
AUTHOR
Gus Wiseman, May 03 2023
STATUS
approved