A362608 Number of integer partitions of n having a unique mode.

0, 1, 2, 2, 4, 5, 7, 11, 16, 21, 29, 43, 54, 78, 102, 131, 175, 233, 295, 389, 490, 623, 794, 1009, 1255, 1579, 1967, 2443, 3016, 3737, 4569, 5627, 6861, 8371, 10171, 12350, 14901, 18025, 21682, 26068, 31225, 37415, 44617, 53258, 63313, 75235, 89173, 105645

Offset: 0

Author

Gus Wiseman, Apr 30 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 partition (3,3,2,1) has greatest multiplicity 2, and a unique part of multiplicity 2 (namely 3), so is counted under a(9).
The a(1) = 1 through a(7) = 11 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)
       (11)  (111)  (22)    (221)    (33)      (322)
                    (211)   (311)    (222)     (331)
                    (1111)  (2111)   (411)     (511)
                            (11111)  (3111)    (2221)
                                     (21111)   (3211)
                                     (111111)  (4111)
                                               (22111)
                                               (31111)
                                               (211111)
                                               (1111111)
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..1000

Crossrefs

For parts instead of multiplicities we have A000041(n-1), ranks A102750.

For median instead of mode we have A238478, complement A238479.

These partitions have ranks A356862.

The complement is counted by A362607, ranks A362605.

For co-mode complement we have A362609, ranks A362606.

For co-mode we have A362610, ranks A359178.

A275870 counts collapsible partitions.

A359893 counts partitions by median.

A362611 counts modes in prime factorization, co-modes A362613.

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

Cf. A002865, A008284, A053263, A098859, A237984, A304442, A327472, A360071, A360687, A362612.

Programs

Mathematica

Table[Length[Select[IntegerPartitions[n],Length[Commonest[#]]==1&]],{n,0,30}]

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

Formula

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