A325269 Number of integer partitions of n with 2 distinct parts or at least 3 parts.

0, 0, 0, 2, 3, 6, 9, 14, 20, 29, 40, 55, 75, 100, 133, 175, 229, 296, 383, 489, 625, 791, 1000, 1254, 1573, 1957, 2434, 3009, 3716, 4564, 5602, 6841, 8347, 10142, 12308, 14882, 17975, 21636, 26013, 31184, 37336, 44582, 53172, 63260, 75173, 89133, 105556

Offset: 0

Author

Gus Wiseman, Apr 18 2019

Keywords

• nonn

Comments

The Heinz numbers of these partitions are given by A080257.

Partitions with 2 distinct parts are in A002133(n). Partitions with at least 3 parts are in A004250(n). Some partitions are in both subsets, so A002133(n)+A004250(n) >= a(n). - R. J. Mathar, Dec 13 2022

Examples

			The a(1) = 1 through a(8) = 20 partitions:
  (21)   (31)    (32)     (42)      (43)       (53)
  (111)  (211)   (41)     (51)      (52)       (62)
         (1111)  (221)    (222)     (61)       (71)
                 (311)    (321)     (322)      (332)
                 (2111)   (411)     (331)      (422)
                 (11111)  (2211)    (421)      (431)
                          (3111)    (511)      (521)
                          (21111)   (2221)     (611)
                          (111111)  (3211)     (2222)
                                    (4111)     (3221)
                                    (22111)    (3311)
                                    (31111)    (4211)
                                    (211111)   (5111)
                                    (1111111)  (22211)
                                               (32111)
                                               (41111)
                                               (221111)
                                               (311111)
                                               (2111111)
                                               (11111111)
		

Crossrefs

Cf. A001221, A001222, A001358, A001399, A007774, A008284, A060687, A080257, A090858, A116608, A325244.

Cf. A000041, A002133, A004250.

Programs

Maple

A325269 := proc(n)
    local a,p,s ;
    a := 0 ;
    for p in combinat[partition](n) do
        s := convert(p,set) ;
        if nops(p) >= 3 or nops(s) = 2 then
            a := a+1 ;
        end if;
    end do:
    a ;
end proc:
seq(A325269(n),n=0..40) ; # R. J. Mathar, Dec 13 2022

Mathematica

Table[Length[Select[IntegerPartitions[n],Length[Union[#]]==2||Length[#]>2&]],{n,0,30}]

Formula

conjecture: a(n) = A000041(n) - A000034(n-1), n>0. - R. J. Mathar, Dec 13 2022