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Heinz numbers of non-self-conjugate integer partitions.
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%I #11 Mar 31 2022 03:03:53

%S 3,4,5,7,8,10,11,12,13,14,15,16,17,18,19,21,22,23,24,25,26,27,28,29,

%T 31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,

%U 54,55,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73

%N Heinz numbers of non-self-conjugate integer partitions.

%C The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions. The sequence lists all Heinz numbers of partitions whose Heinz number is different from that of their conjugate.

%F a(n) != A122111(a(n)).

%e The terms together with their prime indices begin:

%e 3: (2)

%e 4: (1,1)

%e 5: (3)

%e 7: (4)

%e 8: (1,1,1)

%e 10: (3,1)

%e 11: (5)

%e 12: (2,1,1)

%e 13: (6)

%e 14: (4,1)

%e 15: (3,2)

%e 16: (1,1,1,1)

%e 17: (7)

%e 18: (2,2,1)

%e For example, the self-conjugate partition (4,3,3,1) has Heinz number 350, so 350 is not in the sequence.

%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];

%t conj[y_]:=If[Length[y0]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];

%t Select[Range[100],#!=Times@@Prime/@conj[primeMS[#]]&]

%Y The complement is A088902, counted by A000700.

%Y These partitions are counted by A330644.

%Y These are the positions of nonzero terms in A352491.

%Y A000041 counts integer partitions, strict A000009.

%Y A098825 counts permutations by unfixed points.

%Y A238349 counts compositions by fixed points, rank statistic A352512.

%Y A325039 counts partitions w/ same product as conjugate, ranked by A325040.

%Y A352523 counts compositions by unfixed points, rank statistic A352513.

%Y Heinz number (rank) and partition:

%Y - A003963 = product of partition, conjugate A329382

%Y - A008480 = number of permutations of partition, conjugate A321648.

%Y - A056239 = sum of partition

%Y - A122111 = rank of conjugate partition

%Y - A296150 = parts of partition, reverse A112798, conjugate A321649

%Y - A352487 = less than conjugate, counted by A000701

%Y - A352488 = greater than or equal to conjugate, counted by A046682

%Y - A352489 = less than or equal to conjugate, counted by A046682

%Y - A352490 = greater than conjugate, counted by A000701

%Y Cf. A000720, A026424, A120383, A175508, A195017, A238745, A301987, A304360, A316524, A324846, A350841.

%K nonn

%O 1,1

%A _Gus Wiseman_, Mar 20 2022