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The Heinz numbers of the self-conjugate partitions. We define the Heinz number of a partition p = [p_1, p_2, ..., p_r] to be Product(p_j-th prime, j=1...r) (a concept used by _Alois P. Heinz _ in A215366 as an "encoding" of a partition). For example, for the partition [1, 1, 1, 4] we get 2*2*2*7 = 56. It is in the sequence since [1,1,1,4] is self-conjugate. - Emeric Deutsch, Jun 05 2015

A098825 counts permutations by unfixed points.

Cf. A000720, A098825, A195017, A238351, A238352, A238745, A347450, A350841.

- A122111 = rank of conjugate partition.

Cf. A000720, A195017, `A238351, A238352, A238745, `A320324, `A347450, `A350841, `A352523.

350: (4,3,3,1)

416: (6,1,1,1,1,1)

441: (4,4,2,2)

624: (6,2,1,1,1,1)

660: (5,3,2,1,1)

735: (4,4,3,2)

A238352 counts reversed partitions by unfixed points.

A238349 counts comps by fixed points, first col A238351, rank stat A352512.

A352523 counts compositions by unfixed points, rank statistic A352513.

- A003963 = product of partition, conjugate A329382.

- A056239 = sum of partition.

- A122111 = rank of conjugate partition.

- A296150 = parts of partition, reverse A112798, conjugate A321649.

- A352487 = less than conjugate, counted by A000701.

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

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

- A352490 = greater than conjugate, counted by A000701.

Cf. A000720, A195017, `A238351, A238352, A238745, `A320324, `A347450, `A350841, `A352523.

Cf. A056239, A000700, A242422, A215366.

The same count comes from A258116 (h distinct odd parts).

A238349 counts compositions comps by fixed points, first column col A238351, rank statistic stat A352512.

A325039 counts partitions with the same w/ product as their = conjugate, product, ranked by A325040.

Cf. A000720, `~A008292, ~A026424, ~A028260, `~A120383, ~`A175508, A195017, A238745, ~`A304360, `~A320324, ~`A324846, ~`A324847, ~A324850, `~A347450, ~`A350841.

From Gus Wiseman, Jun 28 2022: (Start)

The terms together with their prime indices begin:

1: ()

2: (1)

6: (2,1)

9: (2,2)

20: (3,1,1)

30: (3,2,1)

56: (4,1,1,1)

75: (3,3,2)

84: (4,2,1,1)

125: (3,3,3)

176: (5,1,1,1,1)

210: (4,3,2,1)

264: (5,2,1,1,1)

350: (4,3,3,1)

416: (6,1,1,1,1,1)

441: (4,4,2,2)

624: (6,2,1,1,1,1)

660: (5,3,2,1,1)

735: (4,4,3,2)

(End)

These partitions are counted by A000700.

The same count comes from A258116 (h distinct odd parts).

The complement is A352486, counted by A330644.

These are the positions of zeros in A352491.

A000041 counts integer partitions, strict A000009.

A098825 counts permutations by unfixed points.

A238352 counts reversed partitions by unfixed points.

A238349 counts compositions by fixed points, first column A238351, rank statistic A352512.

A325039 counts partitions with the same product as their conjugate, ranked by A325040.

A352523 counts compositions by unfixed points, rank statistic A352513.

Heinz number (rank) and partition:

- A003963 = product of partition, conjugate A329382

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

- A056239 = sum of partition

- A122111 = rank of conjugate partition

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

- A352487 = less than conjugate, counted by A000701

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

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

- A352490 = greater than conjugate, counted by A000701

Cf. A000720, `~A008292, ~A026424, ~A028260, `~A120383, ~`A175508, A195017, A238745, ~`A304360, `~A320324, ~`A324846, ~`A324847, ~A324850, `~A347450, ~A350841.

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