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A265317 The number of partitions of 2n having segment structure symmetry.

%I A265317 #16 Mar 02 2016 00:04:59
%S A265317 1,1,3,5,10,17,33,53
%N A265317 The number of partitions of 2n having segment structure symmetry.
%C A265317 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 A265317 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 A265317 If the total number of parts is an odd number, then a partition of 2n cannot exhibit segment structure symmetry.
%C A265317 Every partition of n into exactly 2 parts has structure symmetry <1><1>.
%C A265317 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 A265317 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 A265317 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 A265317 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 A265317 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 A265317 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 A265317 Cf. A000041.
%K A265317 nonn,more
%O A265317 0,3
%A A265317 _Gregory L. Simay_, Dec 06 2015