proposed

approved

editing

proposed

In terms of Young partition diagrams, these are partitions whose rectangle from the left (length times minimum) has the same size as the complement.

A097364 counts partitions by (maximum) - (minimum).

A326844 gives the diagram complement size of Heinz partition.

Cf. A237984, A240219, A326836, A326837, A327482, A237755, A237824, A361848, A361852, A349156, A359360, A360068, A360241, A361853, A361906, A361907.

Cf. A237984, A240219, A326836, A326837, A327482, A349156, A359360, A360068, A360241.

`A051293 counts subsets with integer mean.

Cf. A237755, A237824, A361848, A361852, A361853 (ranks A361855), , A361906, A361907.

Cf. A237984, A240219, `A326836, `A326837, `A326846, A327482, `A349156, A359360, A360068, A360241.

Other inequalities: A237755, A237824, A361848, A361852, A361853 (ranks A361855), A361906, A361907.

Cf. A237755, A237824, A361848, A361852, A361853 (ranks A361855), A361906, A361907.

`Cf. A124010, A243503, A268192, A316413, A325351, A325352, A237984, A240219, `A326836, `A326837, `A326846, A355534, A327482, `A349156, A359360, A360068, A360241.

`Cf. A111907, A237984, A240219, A324521, A324562, A327482, A349156, A360068, A360071, A360241, A361394, A361859.

Positions of zeros are A008578.

For median instead of mean we have new.

Other inequalities: A237755, A237824, A361848, A361852, A361853 (ranks A361855), A361906, A361907.

`A051293 counts subsets with integer mean.

`A055396 gives minimum prime index, maximum A061395.

`A112798 list prime indices, sum A056239.

A237755 ptns_len_leq_twimax.

A237824 ptns_max_leq_twi_min.

A326844 gives diagram complement size in the Young diagram of the Heinz partition of n.

A326846 = size of the smallest rectangle containing the prime indices of n.

`A358195 gives Heinz numbers of rows of A358172, even bisection A241916.

A361848 ptns_max_leq_twi_medn.

A361852 ptns_max_less_twi_mean.

A361853 ptns_max_eq_twi_mean, ranks A361855 prix_max_eq_twi_mean.

A361906 ptns_max_geq_twi_mean, unequal case A361907 ptns_max_grtr_twi_mean.

`Cf. A124010, A243503, A268192, A316413, A325351, A325352, A326836, A326837, A326846, A355534, A359360.

In terms of Young diagrams, these are partitions whose maximal rectangle from the left has the same size as the complement.

Both the maximal rectangle from the left and the complement have size 4.

allocated for Gus WisemanNumber of integer partitions of n such that (maximum) - (minimum) = (mean).

0, 0, 0, 1, 0, 1, 0, 3, 2, 2, 0, 7, 0, 3, 6, 10, 0, 13, 0, 17, 10, 5, 0, 40, 12, 6, 18, 34, 0, 62, 0, 50, 24, 8, 60, 125, 0, 9, 32, 169, 0, 165, 0, 95, 176, 11, 0, 373, 114, 198, 54, 143, 0, 384, 254, 574, 66, 14, 0, 1090, 0, 15, 748, 633, 448, 782, 0, 286

1,8

In terms of Young diagrams, these are partitions whose maximal rectangle from the left has the same size as the complement.

The a(4) = 1 through a(12) = 7 partitions:

(31) . (321) . (62) (441) (32221) . (93)

(3221) (522) (33211) (642)

(3311) (4431)

(5322)

(322221)

(332211)

(333111)

The partition y = (4,4,3,1) has maximum 4 and minimum 1 and mean 3, and 4 - 1 = 3, so y is counted under a(12). The diagram of y is:

o o o o

o o o o

o o o .

o . . .

Both the maximal rectangle from the left and the complement have size 4.

Table[Length[Select[IntegerPartitions[n], Max@@#-Min@@#==Mean[#]&]], {n, 30}]

Positions of zeros are 1 and A000040.

Positions of zeros are A008578.

For length instead of mean we have A237832.

For minimum instead of mean we have A118096.

For median instead of mean we have new.

These partitions have ranks A362047.

A000041 counts integer partitions, strict A000009.

A008284 counts partitions by length, A058398 by mean.

A051293 counts subsets with integer mean.

`A055396 gives minimum prime index, maximum A061395.

A067538 counts partitions with integer mean.

`A112798 list prime indices, sum A056239.

A237755 ptns_len_leq_twimax.

A237824 ptns_max_leq_twi_min.

A243055 subtracts the least prime index from the greatest.

A326844 gives complement size in the Young diagram of the Heinz partition of n.

A326846 = size of the smallest rectangle containing the prime indices of n.

`A358195 gives Heinz numbers of rows of A358172, even bisection A241916.

A361848 ptns_max_leq_twi_medn.

A361852 ptns_max_less_twi_mean.

A361853 ptns_max_eq_twi_mean, ranks A361855 prix_max_eq_twi_mean.

A361906 ptns_max_geq_twi_mean, unequal case A361907 ptns_max_grtr_twi_mean.

`Cf. A124010, A243503, A268192, A316413, A325351, A325352, A326836, A326837, A355534, A359360.

`Cf. A111907, A237984, A240219, A324521, A324562, A327482, A349156, A360068, A360071, A360241, A361394, A361859.

allocated

nonn

Gus Wiseman, Apr 10 2023

approved

editing

allocated for Gus Wiseman

allocated

approved