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For not just distinct parts we have : A307683, complement A325347, ranks A359912.

The complement for not just distinct parts is A325347, ranks A359908.

The complement for mean instead of median is A360241, ranks A326621.

A000975 counts subsets with integer median.

A008284 counts partitions by length.

A027193 counts odd-length partitions, strict A067659, ranks A026424.

Cf. A000975, A027193, A090794, A240219, A349156, `A359906, A359907, A360005, ~`A360006, A360071, A360241, A360244, A360245, `~A360557, `A360688.

For not just distinct parts we have A307683, strict A359907, ranks A359908A359912.

The complement for not just distinct parts is A325347, ranks A359912A359908.

For multiplicities instead of distinct parts we have A360690, ranks A360554.

The complement for For multiplicities instead of distinct parts is A360687, we have A360690, ranks A360553A360554.

A027193 counts odd-length partitions, strict A067659, ranks A026424, counting only distinct parts A090794.

A067538 counts partitions with integer mean, strict A102627, ranks A316413.

A304038 lists distinct prime indices, length A001221, sum A066328.

A326619/A326620 gives mean of distinct prime indices.

Cf. A240219 ptns_mean_eq_medn, A327475 subs_int_mean, A359897 strptns_mean_eq_medn, A359906 ptns_int_mean_int_medn, , A360005 prix_medn_times2, A360006 prix_medn_mins, A360071 tet_ptns_len_numdstnct, A360244 ptns_pts_dstnctpts_diff_medn, A360245 ptns_pts_dstnctpts_same_medn, A360457 dstnctprix_medn_times2, A360557 prixdiffs_0prep_nonint_medn, A360688 ptns_0prep_diffs_int_medn.

A360457 gives median of distinct prime indices (times 2).

Cf. A090794, A240219, A349156, `A359906, A359907, A360005, ~`A360006, A360071, A360244, A360245, `~A360557, `A360688.

allocated for Gus WisemanNumber of integer partitions of n whose distinct parts have non-integer median.

0, 0, 1, 1, 4, 3, 8, 6, 13, 11, 21, 17, 34, 36, 55, 61, 97, 115, 162, 191, 270, 328, 427, 514, 666, 810, 1027, 1211, 1530, 1832, 2260, 2688, 3342, 3952, 4824, 5746, 7010, 8313, 10116, 11915, 14436, 17074, 20536, 24239, 29053, 34170, 40747, 47865, 56830, 66621

1,5

The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).

The a(1) = 0 through a(9) = 13 partitions:

. . (21) (211) (32) (411) (43) (332) (54)

(41) (2211) (52) (611) (63)

(221) (21111) (61) (22211) (72)

(2111) (322) (41111) (81)

(2221) (221111) (441)

(4111) (2111111) (522)

(22111) (3222)

(211111) (6111)

(22221)

(222111)

(411111)

(2211111)

(21111111)

For example, the partition y = (5,3,3,2,1,1) has distinct parts {1,2,3,5}, with median 5/2, so y is counted under a(15).

Table[Length[Select[IntegerPartitions[n], !IntegerQ[Median[Union[#]]]&]], {n, 30}]

For not just distinct parts we have A307683, strict A359907, ranks A359908.

The complement for not just distinct parts is A325347, ranks A359912.

The complement for mean instead of median is A360241, ranks A326621.

These partitions have ranks A360551.

For multiplicities instead of distinct parts we have A360690, ranks A360554.

The complement is counted by A360686, strict A359907, ranks A360550.

The complement for multiplicities instead of distinct parts is A360687, ranks A360553.

A000041 counts integer partitions, strict A000009.

A000975 counts subsets with integer median.

A008284 counts partitions by length.

A027193 counts odd-length partitions, strict A067659, ranks A026424, counting only distinct parts A090794.

A067538 counts partitions with integer mean, strict A102627, ranks A316413.

A116608 counts partitions by number of distinct parts.

A304038 lists distinct prime indices, length A001221, sum A066328.

A326619/A326620 gives mean of distinct prime indices.

A359893 and A359901 count partitions by median, odd-length A359902.

Cf. A240219 ptns_mean_eq_medn, A327475 subs_int_mean, A359897 strptns_mean_eq_medn, A359906 ptns_int_mean_int_medn, , A360005 prix_medn_times2, A360006 prix_medn_mins, A360071 tet_ptns_len_numdstnct, A360244 ptns_pts_dstnctpts_diff_medn, A360245 ptns_pts_dstnctpts_same_medn, A360457 dstnctprix_medn_times2, A360557 prixdiffs_0prep_nonint_medn, A360688 ptns_0prep_diffs_int_medn.

allocated

nonn

Gus Wiseman, Feb 22 2023

approved

editing

allocated for Gus Wiseman

allocated

approved