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Number of unordered pairs of strict integer partitions of n.
2

%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