A359906 -id:A359906 - cp's OEIS frontend

A359898 Number of strict integer partitions of n whose parts do not have the same mean as median.

0, 0, 0, 0, 0, 0, 0, 1, 2, 1, 4, 6, 5, 11, 12, 14, 21, 29, 26, 44, 44, 58, 68, 92, 92, 118, 137, 165, 192, 241, 223, 324, 353, 405, 467, 518, 594, 741, 809, 911, 987, 1239, 1276, 1588, 1741, 1823, 2226, 2566, 2727, 3138, 3413, 3905, 4450, 5093, 5434, 6134

Offset: 0

Gus Wiseman, Jan 20 2023

nonn

			The a(7) = 1 through a(13) = 11 partitions:
  (4,2,1)  (4,3,1)  (6,2,1)  (5,3,2)  (5,4,2)    (6,5,1)    (6,4,3)
           (5,2,1)           (5,4,1)  (6,3,2)    (7,3,2)    (6,5,2)
                             (6,3,1)  (6,4,1)    (8,3,1)    (7,4,2)
                             (7,2,1)  (7,3,1)    (9,2,1)    (7,5,1)
                                      (8,2,1)    (6,3,2,1)  (8,3,2)
                                      (5,3,2,1)             (8,4,1)
                                                            (9,3,1)
                                                            (10,2,1)
                                                            (5,4,3,1)
                                                            (6,4,2,1)
                                                            (7,3,2,1)

The non-strict version is ranked by A359890, complement A359889.
The non-strict version is A359894, complement A240219.
The complement is counted by A359897.
The odd-length case is A359900, complement A359899.
A000041 counts partitions, strict A000009.
A008284/A058398/A327482 count partitions by mean, ranked by A326567/A326568.
A008289 counts strict partitions by mean.
A067538 counts ptns with integer mean, strict A102627, ranked by A316413.
A237984 counts ptns containing their mean, strict A240850, ranked by A327473.
A325347 counts ptns with integer median, strict A359907, ranked by A359908.
A359893 and A359901 count partitions by median, odd-length A359902.
Cf. A000016, A065795, A066571, A082550, A135342, A240851, A327475, A327482, A328966, A359906.

Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&Mean[#]!=Median[#]&]],{n,0,30}]

A360686 Number of integer partitions of n whose distinct parts have integer median.

Original entry on oeis.org

1, 2, 2, 4, 3, 8, 7, 16, 17, 31, 35, 60, 67, 99, 121, 170, 200, 270, 328, 436, 522, 674, 828, 1061, 1292, 1626, 1983, 2507, 3035, 3772, 4582, 5661, 6801, 8358, 10059, 12231, 14627, 17702, 21069, 25423, 30147, 36100, 42725, 50936, 60081, 71388, 84007, 99408

Offset: 1

Gus Wiseman, Feb 20 2023

nonn

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) = 1 through a(8) = 16 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (111)  (22)    (311)    (33)      (331)      (44)
                    (31)    (11111)  (42)      (421)      (53)
                    (1111)           (51)      (511)      (62)
                                     (222)     (3211)     (71)
                                     (321)     (31111)    (422)
                                     (3111)    (1111111)  (431)
                                     (111111)             (521)
                                                          (2222)
                                                          (3221)
                                                          (3311)
                                                          (4211)
                                                          (5111)
                                                          (32111)
                                                          (311111)
                                                          (11111111)
For example, the partition y = (7,4,2,1,1) has distinct parts {1,2,4,7} with median 3, so y is counted under a(15).

For all parts: A325347, strict A359907, ranks A359908, complement A307683.
For mean instead of median: A360241, ranks A326621.
These partitions have ranks A360550, complement A360551.
For multiplicities instead of distinct parts: A360687.
The complement is counted by A360689.
A000041 counts integer partitions, strict A000009.
A000975 counts subsets with integer median.
A027193 counts odd-length partitions, strict A067659, ranks A026424.
A067538 counts partitions with integer mean, strict A102627, ranks A316413.
A116608 counts partitions by number of distinct parts.
A359893 and A359901 count partitions by median, odd-length A359902.
Cf. A240219, A359906, A360005, A360071, A360244, A360245, A360556, A360688.

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

A359911 Number of integer factorizations of n into factors > 1 without the same mean as median.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 2, 0, 0, 0, 1, 0, 1, 0, 3, 0, 0, 0, 3, 0, 0, 0, 3, 0, 1, 0, 1, 1, 0, 0, 6, 0, 1, 0, 1, 0, 3, 0, 3, 0, 0, 0, 4, 0, 0, 1, 4, 0, 1, 0, 1, 0, 1, 0, 9, 0, 0, 1, 1, 0, 1, 0, 6, 1, 0, 0, 5, 0, 0, 0, 3, 0, 5, 0, 1, 0, 0, 0, 13, 0, 1, 1, 3, 0, 1, 0, 3, 0, 0, 0, 10

Offset: 1

Gus Wiseman, Jan 24 2023

nonn

			The a(72) = 9 factorizations: (2*2*2*3*3), (2*2*2*9), (2*2*3*6), (2*2*18), (2*3*12), (2*4*9), (2*6*6), (3*3*8), (3*4*6).

Antti Karttunen, Table of n, a(n) for n = 1..65537

The version for partitions is A359894, complement A240219.
The complement is counted by A359909, odd-length A359910.
A001055 counts factorizations.
A326622 counts factorizations with integer mean, strict A328966.
Cf. A316313, A326567/A326568, A359897, A359889, A359906, A360005.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[Select[facs[n],Mean[#]!=Median[#]&]],{n,100}]

median(lista) = if((#lista)%2, lista[(1+#lista)/2], (lista[#lista/2]+lista[1+(#lista/2)])/2);
A359911(n, m=n, facs=List([])) = if(1==n, (#facs>0 && (median(facs)!=(vecsum(Vec(facs))/#facs))), my(s=0, newfacs); fordiv(n, d, if((d>1)&&(d<=m), newfacs = List(facs); listput(newfacs,d); s += A359911(n/d, d, newfacs))); (s)); \\ Antti Karttunen, Jan 20 2025

For n > 1, a(n) = A001055(n) - A359909(n). - Antti Karttunen, Jan 20 2025

Data section extended to a(108) by Antti Karttunen, Jan 20 2025

cp's OEIS Frontend

A359898 Number of strict integer partitions of n whose parts do not have the same mean as median.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

A360686 Number of integer partitions of n whose distinct parts have integer median.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A359911 Number of integer factorizations of n into factors > 1 without the same mean as median.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions