%I
%S 1,1,3,5,10,17,33,53
%N The number of partitions of 2n having segment structure symmetry.
%C Define a segmented partition a(n, k, <s(1)..s(j)>) to be a partition of n with exactly k parts, with s(j) parts t(j) identical to each other and distinct from all other parts. Note that n>=k, j<=k, 0<=s(j)<=k, s(1)t(1) + ... + s(j)t(j) = n and s(1) + ... + s(j) = k.
%C A partition of 2n has segment structure symmetry if there is at least one way it can be arranged into two consecutive sequences (no central summand), with each sequence exhibiting the same segment structure.
%C If the total number of parts is an odd number, then a partition of 2n cannot exhibit segment structure symmetry.
%C Every partition of n into exactly 2 parts has structure symmetry <1><1>.
%C Except for the case of partitions with a segment structure of <3,1>, every partition of n into exactly 4 parts has segment structure symmetry.
%C Except for the case of partitions with a segment structure of <5,1> or <3,1,1>, every partition of 2n into exactly 6 integers has segment structure symmetry.
%e The partition 5,3,3,1,1,1 can be rearranged into two consecutive sequences (separated by a / for clarity) 1,1,5/3,3,1 which exhibits the segment structure symmetry <2,1><2,1>, and so counts as one of the 53 partitions of 14 exhibiting this symmetry.
%e The partitions of 14 with exactly 2 parts are 13,1; 12,2; 11,3; 10,4; 9,5; 8,6; and 7,7. All of them, including 7,7, exhibit segment structure symmetry in the form of <1><1>.
%e The partition 5,5,3,1 is an example of a partition of 14 with exactly 4 parts with segment structure <2,1,1>. This partition can be arranged into two consecutive sequences 5,3/5,1, which exhibits the segment structure symmetry, <1,1><1,1>.
%e The partition 3,3,3,2,2,1 is an example of a partition of 14 with exactly 6 parts with segment structure <3,2,1>. This partition can be arranged into two consecutive sequences 3,3,1/2,2,3, which exhibits the segment structure symmetry <2,1><2,1>.
%Y Cf. A000041.
%K nonn,more
%O 0,3
%A _Gregory L. Simay_, Dec 06 2015
