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

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

Author

Gus Wiseman, May 03 2023

Keywords

• nonn

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}.

Examples

			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
		

Links

• John Tyler Rascoe, Table of n, a(n) for n = 0..500

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.

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

Programs

Mathematica

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

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

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