A090794 -id:A090794 - cp's OEIS frontend

A360687 Number of integer partitions of n whose multiplicities have integer median.

1, 2, 3, 4, 5, 9, 10, 16, 22, 34, 42, 65, 80, 115, 145, 195, 240, 324, 396, 519, 635, 814, 994, 1270, 1549, 1952, 2378, 2997, 3623, 4521, 5466, 6764, 8139, 10008, 12023, 14673, 17534, 21273, 25336, 30593, 36302, 43575, 51555, 61570, 72653, 86382, 101676

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)  (21)   (22)    (32)     (33)      (43)       (44)
             (111)  (31)    (41)     (42)      (52)       (53)
                    (1111)  (2111)   (51)      (61)       (62)
                            (11111)  (222)     (421)      (71)
                                     (321)     (2221)     (431)
                                     (2211)    (3211)     (521)
                                     (3111)    (4111)     (2222)
                                     (111111)  (211111)   (3221)
                                               (1111111)  (3311)
                                                          (4211)
                                                          (5111)
                                                          (32111)
                                                          (221111)
                                                          (311111)
                                                          (11111111)
For example, the partition y = (3,2,2,1) has multiplicities (1,2,1), and the multiset {1,1,2} has median 1, so y is counted under a(8).

The case of an odd number of multiplicities is A090794.
For mean instead of median we have A360069, ranks A067340.
These partitions have ranks A360553.
The complement is counted by A360690, ranks A360554.
A058398 counts partitions by mean, see also A008284, A327482.
A124010 gives prime signature, sorted A118914, mean A088529/A088530.
A325347 = partitions w/ integer median, strict A359907, complement A307683.
A359893 and A359901 count partitions by median, odd-length A359902.
Cf. A000975, A329976, A359908, A360068, A360460, A360550, A360556, A360688.

Table[Length[Select[IntegerPartitions[n],IntegerQ[Median[Length/@Split[#]]]&]],{n,30}]

A104575 Alternating sum of diagonals in A060177.

Original entry on oeis.org

1, -1, -2, -1, -1, 3, 1, 7, 4, 4, 4, 2, -9, -7, -7, -28, -17, -25, -15, -24, -11, -8, 34, 19, 53, 46, 108, 110, 106, 113, 122, 108, 75, 103, -16, -87, -107, -169, -329, -257, -574, -501, -676, -609, -749, -588, -808, -548, -521, -315, -240, 369, 485, 865, 1099, 1738, 2129, 2686, 3088, 3460, 4103, 4011, 4480, 3983

Offset: 0

Vladeta Jovovic, Apr 21 2005

A090794(n) = (A000041(n)-a(n))/2. A092306(n) = (A000041(n)+a(n))/2.

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Convolution inverse of A006951.

Cf. A000041, A008965, A081362, A092306, A268498.

CoefficientList[Series[Product[(1-2x^k)/(1-x^k),{k,70}],{x,0,70}],x] (* Harvey P. Dale, Jan 21 2021 *)

N=66; x='x+O('x^N); Vec(prod(k=1, N, 1-x^k/(1-x^k))) \\ Seiichi Manyama, Oct 05 2019

G.f.: Product_{i>0} (1 - 2*x^i)/(1 - x^i).

Euler transform of -A008965(n).

a(0)=1 prepended by Seiichi Manyama, Oct 05 2019

A092306 Number of partitions of n such that the set of parts has an even number of elements.

Original entry on oeis.org

1, 0, 0, 1, 2, 5, 6, 11, 13, 17, 23, 29, 34, 47, 64, 74, 107, 136, 185, 233, 308, 392, 518, 637, 814, 1002, 1272, 1560, 1912, 2339, 2863, 3475, 4212, 5123, 6147, 7398, 8935, 10734, 12843, 15464, 18382, 22041, 26249, 31326, 37213, 44273, 52375, 62103, 73376

Offset: 0

Vladeta Jovovic, Feb 12 2004

			The partitions of five are: {{5}, {4, 1}, {3, 2}, {3, 1, 1}, {2, 2, 1}, {2, 1, 1, 1}, {1, 1, 1, 1, 1}}, The seven partitions have 1, 2, 2, 2, 2, 2 and 1 distinct parts respectively.
n=6 has A000041(6)=11 partitions: 6, 5+1, 4+2, 4+1+1, 3+3, 3+2+1, 3+1+1+1, 2+2+2, 2+2+1+1, 2+1+1+1+1 and 1+1+1+1+1+1 with partition sets: {6}, {1,5}, {2,4}, {1,4}, {3}, {1,2,3}, {1,3}, {2}, {1,2}, {1,2} and {1}, six of them have an even number of elements, therefore a(6)=6.

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A060177, A002133, A027187, A090794.

import Data.List (group)
a092306 = length . filter even . map (length . group) . ps 1 where
   ps x 0 = [[]]
   ps x y = [t:ts | t <- [x..y], ts <- ps t (y - t)]
-- Reinhard Zumkeller, Dec 19 2013

b:= proc(n, i, t) option remember; `if`(n=0, t, `if`(i<1, 0,
       b(n, i-1, t) +add(b(n-i*j, i-1, 1-t), j=1..n/i)))
    end:
a:= n-> b(n, n, 1):
seq(a(n), n=0..50);  # Alois P. Heinz, Jan 29 2014

f[n_] := Count[ Mod[ Length /@ Union /@ IntegerPartitions[n], 2], 0]; Table[ f[n], {n, 0, 49}] (* Robert G. Wilson v, Feb 16 2004, updated by Jean-François Alcover, Jan 29 2014 *)

a(n) = b(n, 1, 0, 1) with b(n, i, j, f) = if iReinhard Zumkeller, Feb 19 2004
G.f.: F(x)*G(x)/2, where F(x) = 1+Product(1-2*x^i, i=1..infinity) and G(x) = 1/Product(1-x^i, i=1..infinity).
a(n) = (A000041(n)+A104575(n))/2.
G.f. A(x) equals the main diagonal entries in the 2 X 2 matrix Product_{n >= 1} [1, x^n/(1 - x^n); x^n/(1 - x^n), 1] = [A(x), B(x); B(x), A(x)], where B(x) is the g.f. of A090794. - Peter Bala, Feb 10 2021

More terms from Robert G. Wilson v, Feb 16 2004

A261555 Triangle read by rows: T(n,k) is number of partitions of n having at least k distinct parts (n >= 1, k >= 1).

Original entry on oeis.org

1, 2, 3, 1, 5, 2, 7, 5, 11, 7, 1, 15, 13, 2, 22, 18, 5, 30, 27, 10, 42, 38, 16, 1, 56, 54, 27, 2, 77, 71, 42, 5, 101, 99, 62, 10, 135, 131, 87, 20, 176, 172, 128, 31, 1, 231, 226, 171, 54, 2, 297, 295, 236, 82, 5, 385, 379, 311, 127, 10, 490, 488, 417, 182, 20

Offset: 1

Michel Marcus, Aug 24 2015

From Omar E. Pol, Sep 14 2016: (Start)
Row n has length A003056(n) hence the first element of column k is in row A000217(k).
Row sums give A000070.
Alternating row sums give A090794.
Column 1 is A000041, n >= 1. (End)
[0, 0] together with column 2 gives A144300. - Omar E. Pol, Sep 17 2016

			Triangle starts:
1;
2;
3,  1;
5,  2;
7,  5;
11, 7,  1;
15, 13, 2;
22, 18, 5;
30, 27, 10;
42, 38, 16, 1;
56, 54, 27, 2;
77, 71, 42, 5;
...

Jacques Barbot, Essai sur la structuration de l'analyse combinatoire, Paris, Dulac, 1973, Annexe 12A, p. 74.

Alois P. Heinz, Rows n = 1..500, flattened

Cf. A000005, A000041, A000070, A000217, A003056, A060177, A090794, A116608, A144300.

Table[DeleteCases[Map[Count[Map[Length@ Union@ # &, IntegerPartitions@ n], k_ /; k >= #] &, Range@ n], 0], {n, 19}] // Flatten (* Michael De Vlieger, Sep 14 2016 *)

T(n,k) = Sum_{j>=k} A116608(n,j) assuming A116608(n,j)=0 when j>A003056(n).

T(n,1) - T(n,2) = A000005(n). - Omar E. Pol, Sep 17 2016

More terms from Alois P. Heinz, Aug 24 2015

A360690 Number of integer partitions of n with non-integer median of multiplicities.

Original entry on oeis.org

0, 0, 0, 1, 2, 2, 5, 6, 8, 8, 14, 12, 21, 20, 31, 36, 57, 61, 94, 108, 157, 188, 261, 305, 409, 484, 632, 721, 942, 1083, 1376, 1585, 2004, 2302, 2860, 3304, 4103, 4742, 5849, 6745, 8281, 9599, 11706, 13605, 16481, 19176, 23078, 26838, 32145, 37387, 44465

Offset: 1

Gus Wiseman, Feb 22 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) = 0 through a(9) = 8 partitions:
  .  .  .  (211)  (221)  (411)    (322)    (332)      (441)
                  (311)  (21111)  (331)    (422)      (522)
                                  (511)    (611)      (711)
                                  (22111)  (22211)    (22221)
                                  (31111)  (41111)    (33111)
                                           (2111111)  (51111)
                                                      (2211111)
                                                      (3111111)
For example, the partition y = (3,2,2,1) has multiplicities (1,2,1), and the multiset {1,1,2} has median 1, so y is not counted under a(8).

These partitions have ranks A360554.
The complement is counted by A360687, ranks A360553.
A058398 counts partitions by mean, see also A008284, A327482.
A124010 gives prime signature, sorted A118914, mean A088529/A088530.
A325347 = partitions w/ integer median, strict A359907, complement A307683.
A359893 and A359901 count partitions by median, odd-length A359902.
A360069 = partitions with integer mean of multiplicities, ranks A067340.
Cf. A000041, A000975, A090794, A329976, A359908, A360068, A360460, A360550, A360556, A360688.

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

A360689 Number of integer partitions of n whose distinct parts have non-integer median.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Feb 22 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) = 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).

For not just distinct parts: A307683, complement A325347, ranks A359912.
These partitions have ranks A360551.
The complement is counted by A360686, strict A359907, ranks A360550.
For multiplicities instead of distinct parts we have A360690, ranks A360554.
A000041 counts integer partitions, strict A000009.
A116608 counts partitions by number of distinct parts.
A359893 and A359901 count partitions by median, odd-length A359902.
A360457 gives median of distinct prime indices (times 2).
Cf. A000975, A027193, A090794, A240219, A349156, A360005, A360071, A360241, A360244, A360245.

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

cp's OEIS Frontend

A360687 Number of integer partitions of n whose multiplicities have integer median.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A104575 Alternating sum of diagonals in A060177.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A092306 Number of partitions of n such that the set of parts has an even number of elements.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

Formula

Extensions

A261555 Triangle read by rows: T(n,k) is number of partitions of n having at least k distinct parts (n >= 1, k >= 1).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A360690 Number of integer partitions of n with non-integer median of multiplicities.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A360689 Number of integer partitions of n whose distinct parts have non-integer median.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica