A349801 - OEIS

A349801

Number of integer partitions of n into three or more parts or into two equal parts.

%I #6 Dec 25 2021 02:44:58

%S 0,0,1,1,3,4,8,11,18,25,37,50,71,94,128,168,223,288,376,480,617,781,

%T 991,1243,1563,1945,2423,2996,3704,4550,5589,6826,8333,10126,12293,

%U 14865,17959,21618,25996,31165,37318,44562,53153,63239,75153,89111,105535,124730

%N Number of integer partitions of n into three or more parts or into two equal parts.

%C This sequence arose as the following degenerate case. If we define a sequence to be alternating if it is alternately strictly increasing and strictly decreasing, starting with either, then a(n) is the number of non-alternating integer partitions of n. Under this interpretation:

%C - The non-strict case is A047967, weak A349796, weak complement A349795.

%C - The complement is counted by A065033(n) = ceiling(n/2) for n > 0.

%C - These partitions are ranked by A289553 \ {1}, complement A167171 \/ {1}.

%C - The version for compositions is A345192, ranked by A345168.

%C - The weak version for compositions is A349053, ranked by A349057.

%C - The weak version is A349061, complement A349060, ranked by A349794.

%F a(1) = 0; a(n > 0) = A000041(n) - ceiling(n/2).

%e The a(2) = 1 through a(7) = 11 partitions:

%e (11) (111) (22) (221) (33) (322)

%e (211) (311) (222) (331)

%e (1111) (2111) (321) (421)

%e (11111) (411) (511)

%e (2211) (2221)

%e (3111) (3211)

%e (21111) (4111)

%e (111111) (22111)

%e (31111)

%e (211111)

%e (1111111)

%t Table[Length[Select[IntegerPartitions[n],MatchQ[#,{x_,x_}|{_,_,__}]&]],{n,0,10}]

%Y A000041 counts partitions, ordered A011782.

%Y A001250 counts alternating permutations, complement A348615.

%Y A004250 counts partitions into three or more parts, strict A347548.

%Y A025047/A025048/A025049 count alternating compositions, ranked by A345167.

%Y A096441 counts weakly alternating 0-appended partitions.

%Y A345165 counts partitions w/ no alternating permutation, complement A345170.

%Y Cf. A000070, A001700, A002865, A117298, A117989, A102726, A128761, A345162, A345163, A345166, A349798.

%K nonn

%O 0,5

%A _Gus Wiseman_, Dec 23 2021