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%I #9 Oct 09 2023 12:58:23
%S 1,1,1,3,3,6,10,15,21,36,55,78,120,171,253,378,528,741,1081,1485,2080,
%T 2926,4005,5460,7503,10153,13695,18528,24753,32896,43956,57970,76245,
%U 100576,131328,171405,223446,289180,373680,482653,619941,794430,1017451,1296855
%N Number of unordered pairs of strict integer partitions of n.
%F a(n) = A000217(A000009(n)).
%F Composition of A000009 and A000217.
%e The a(1) = 1 through a(7) = 15 unordered pairs of strict partitions:
%e {1,1} {2,2} {3,3} {4,4} {5,5} {6,6} {7,7}
%e {3,21} {4,31} {5,32} {6,42} {7,43}
%e {21,21} {31,31} {5,41} {6,51} {7,52}
%e {32,32} {42,42} {7,61}
%e {32,41} {42,51} {43,43}
%e {41,41} {51,51} {43,52}
%e {6,321} {43,61}
%e {42,321} {52,52}
%e {51,321} {52,61}
%e {321,321} {61,61}
%e {7,421}
%e {43,421}
%e {52,421}
%e {61,421}
%e {421,421}
%t Table[Length[Select[Tuples[Select[IntegerPartitions[n], UnsameQ@@#&],2],OrderedQ]],{n,0,30}]
%Y For non-strict partitions we have A086737.
%Y The disjoint case is A108796, non-strict A260669.
%Y The ordered version is A304990, disjoint A032302.
%Y The ordered disjoint case is A365662.
%Y Excluding constant pairs gives A366132.
%Y A000041 counts integer partitions, strict A000009.
%Y A002219 and A237258 count partitions of 2n including a partition of n.
%Y A364272 counts sum-full strict partitions, sum-free A364349.
%Y Cf. A000712, A007582, A054440, A064914, A260664.
%K nonn
%O 0,4
%A _Gus Wiseman_, Oct 08 2023