A367410 -id:A367410 - cp's OEIS frontend

A367395 Number of strict integer partitions of n whose length is the sum of two distinct parts.

0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 2, 2, 2, 3, 3, 5, 5, 7, 8, 11, 13, 17, 19, 25, 28, 35, 41, 49, 57, 68, 78, 92, 107, 124, 143, 166, 192, 220, 254, 291, 335, 382, 439, 499, 572, 649, 741, 840, 956, 1080, 1226, 1383, 1566, 1762, 1988, 2235, 2515, 2822, 3166, 3547

Offset: 0

Gus Wiseman, Nov 19 2023

nonn

			The strict partition (5,3,2,1) has 4 = 3 + 1 so is counted under a(11).
The a(6) = 1 through a(17) = 7 strict partitions (A..E = 10..14):
  321  421  521  621  721   821   921   A21   B21   C21    D21    E21
                      4321  5321  6321  5431  6431  6531   7531   7631
                                        7321  8321  7431   8431   8531
                                                    9321   A321   9431
                                                    54321  64321  B321
                                                                  65321
                                                                  74321

The following sequences count and rank integer partitions and finite sets according to whether their length is a subset-sum, linear combination, or semi-sum of the parts. The current sequence is starred.
             sum-full  sum-free  comb-full comb-free semi-full semi-free
            -----------------------------------------------------------
partitions:  A367212   A367213   A367218   A367219   A367394   A367398
strict:      A367214   A367215   A367220   A367221   A367395*  A367399
subsets:     A367216   A367217   A367222   A367223   A367396   A367400
ranks:       A367224   A367225   A367226   A367227   A367397   A367401
A000041 counts partitions, strict A000009.
A002865 counts partitions whose length is a part, complement A229816.
A088809/A093971 count twofold sum-full subsets.
A236912 counts partitions containing no semi-sum, ranks A364461.
A237113 counts partitions containing a semi-sum, ranks A364462.
A237668 counts sum-full partitions, sum-free A237667.
A366738 counts semi-sums of partitions, strict A366741.
Triangles:
A008284 counts partitions by length, strict A008289.
A365541 counts subsets with a semi-sum k.
A367404 counts partitions with a semi-sum k, strict A367405.
Cf. A000700, A238628, A363225, A364272, A364534, A365661, A365925, A367410, A367411.

Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&MemberQ[Total/@Subsets[#,{2}], Length[#]]&]], {n,0,30}]

A367399 Number of strict integer partitions of n whose length is not the sum of any two distinct parts.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 3, 4, 5, 7, 8, 10, 13, 15, 19, 22, 27, 31, 38, 43, 51, 59, 70, 79, 94, 107, 124, 143, 165, 188, 218, 248, 283, 324, 369, 419, 476, 540, 610, 691, 778, 878, 987, 1111, 1244, 1399, 1563, 1750, 1954, 2184, 2432, 2714, 3016, 3358, 3730, 4143

Offset: 0

Gus Wiseman, Nov 19 2023

nonn

			The strict partition y = (6,4,2,1) has semi-sums {3,5,6,7,8,10}, which do not include 4, so y is counted under a(13).
The a(6) = 3 through a(13) = 15 strict partitions:
  (6)    (7)    (8)      (9)      (10)     (11)     (12)       (13)
  (4,2)  (4,3)  (5,3)    (5,4)    (6,4)    (6,5)    (7,5)      (7,6)
  (5,1)  (5,2)  (6,2)    (6,3)    (7,3)    (7,4)    (8,4)      (8,5)
         (6,1)  (7,1)    (7,2)    (8,2)    (8,3)    (9,3)      (9,4)
                (4,3,1)  (8,1)    (9,1)    (9,2)    (10,2)     (10,3)
                         (4,3,2)  (5,3,2)  (10,1)   (11,1)     (11,2)
                         (5,3,1)  (5,4,1)  (5,4,2)  (5,4,3)    (12,1)
                                  (6,3,1)  (6,3,2)  (6,4,2)    (6,4,3)
                                           (6,4,1)  (6,5,1)    (6,5,2)
                                           (7,3,1)  (7,3,2)    (7,4,2)
                                                    (7,4,1)    (7,5,1)
                                                    (8,3,1)    (8,3,2)
                                                    (5,4,2,1)  (8,4,1)
                                                               (9,3,1)
                                                               (6,4,2,1)

The following sequences count and rank integer partitions and finite sets according to whether their length is a subset-sum, linear combination, or semi-sum of the parts. The current sequence is starred.
             sum-full  sum-free  comb-full comb-free semi-full semi-free
            -----------------------------------------------------------
partitions:  A367212   A367213   A367218   A367219   A367394   A367398
strict:      A367214   A367215   A367220   A367221   A367395   A367399*
subsets:     A367216   A367217   A367222   A367223   A367396   A367400
ranks:       A367224   A367225   A367226   A367227   A367397   A367401
A000041 counts partitions, strict A000009.
A002865 counts partitions whose length is a part, complement A229816.
A365924 counts incomplete partitions, strict A365831.
A236912 counts partitions with no semi-sum of the parts, ranks A364461.
A237667 counts sum-free partitions, sum-full A237668.
A366738 counts semi-sums of partitions, strict A366741.
A367403 counts partitions without covering semi-sums, strict A367411.
Triangles:
A008284 counts partitions by length, strict A008289.
A365541 counts subsets with a semi-sum k.
A367404 counts partitions with a semi-sum k, strict A367405.
Cf. A000700, A126796, A188431, A238628, A237113, A363225, A364272, A365658, A365918, A365925, A367410.

Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&FreeQ[Total/@Subsets[#,{2}], Length[#]]&]], {n,0,15}]

A367402 Number of integer partitions of n whose semi-sums cover an interval of positive integers.

Original entry on oeis.org

1, 1, 2, 3, 5, 6, 9, 10, 13, 17, 20, 26, 31, 38, 44, 58, 64, 81, 95, 116, 137, 166, 192, 233, 278, 330, 385, 459, 542, 636, 759, 879, 1038, 1211, 1418, 1656, 1942, 2242, 2618, 3029, 3535, 4060, 4735, 5429, 6299, 7231, 8346, 9556, 11031, 12593, 14482, 16525

Offset: 0

Gus Wiseman, Nov 17 2023

nonn

We define a semi-sum of a multiset to be any sum of a 2-element submultiset. This is different from sums of pairs of elements. For example, 2 is the sum of a pair of elements of {1}, but there are no semi-sums.

			The partition y = (3,2,1,1) has semi-sums {2,3,4,5}, which is an interval, so y is counted under a(7).
The a(1) = 1 through a(8) = 13 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (21)   (22)    (32)     (33)      (43)       (44)
             (111)  (31)    (41)     (42)      (52)       (53)
                    (211)   (221)    (51)      (61)       (62)
                    (1111)  (2111)   (222)     (322)      (71)
                            (11111)  (321)     (2221)     (332)
                                     (2211)    (3211)     (2222)
                                     (21111)   (22111)    (3221)
                                     (111111)  (211111)   (22211)
                                               (1111111)  (32111)
                                                          (221111)
                                                          (2111111)
                                                          (11111111)

For parts instead of sums we have A034296, ranks A073491.
For all subset-sums we have A126796, ranks A325781, strict A188431.
The complement for parts instead of sums is A239955, ranks A073492.
The complement for all sub-sums is A365924, ranks A365830, strict A365831.
The complement is counted by A367403.
The strict case is A367410, complement A367411.
A000009 counts partitions covering an initial interval, ranks A055932.
A086971 counts semi-sums of prime indices.
A261036 counts complete partitions by maximum.
A276024 counts positive subset-sums of partitions, strict A284640.
Cf. A000041, A002033, A046663, A108917, A264401, A304792, A365543, A365661, A365918, A365921.

Table[Length[Select[IntegerPartitions[n], (d=Total/@Subsets[#,{2}];If[d=={}, {}, Range[Min@@d,Max@@d]]==Union[d])&]], {n,0,15}]

A367403 Number of integer partitions of n whose semi-sums do not cover an interval of positive integers.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 2, 5, 9, 13, 22, 30, 46, 63, 91, 118, 167, 216, 290, 374, 490, 626, 810, 1022, 1297, 1628, 2051, 2551, 3176, 3929, 4845, 5963, 7311, 8932, 10892, 13227, 16035, 19395, 23397, 28156, 33803, 40523, 48439, 57832, 68876, 81903, 97212, 115198

Offset: 0

Gus Wiseman, Nov 17 2023

nonn

			The a(0) = 0 through a(9) = 13 partitions:
  .  .  .  .  .  (311)  (411)   (331)    (422)     (441)
                        (3111)  (421)    (431)     (522)
                                (511)    (521)     (531)
                                (4111)   (611)     (621)
                                (31111)  (3311)    (711)
                                         (4211)    (4311)
                                         (5111)    (5211)
                                         (41111)   (6111)
                                         (311111)  (33111)
                                                   (42111)
                                                   (51111)
                                                   (411111)
                                                   (3111111)

The complement for parts instead of sums is A034296, ranks A073491.
The complement for all sub-sums is A126796, ranks A325781, strict A188431.
For parts instead of sums we have A239955, ranks A073492.
For all subset-sums we have A365924, ranks A365830, strict A365831.
The complement is counted by A367402.
The strict case is A367411, complement A367410.
A000009 counts partitions covering an initial interval, ranks A055932.
A086971 counts semi-sums of prime indices.
A261036 counts complete partitions by maximum.
A276024 counts positive subset-sums of partitions, strict A284640.
Cf. A000041, A002033, A046663, A108917, A264401, A304792, A365543, A365658.

Table[Length[Select[IntegerPartitions[n], (d=Total/@Subsets[#,{2}];If[d=={}, {}, Range[Min@@d,Max@@d]]!=Union[d])&]], {n,0,15}]

A367411 Number of strict integer partitions of n whose semi-sums do not cover an interval of positive integers.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 2, 2, 4, 5, 8, 10, 14, 16, 23, 27, 35, 42, 52, 61, 75, 89, 106, 126, 149, 173, 204, 237, 274, 319, 369, 424, 490, 560, 642, 734, 838, 952, 1085, 1231, 1394, 1579, 1784, 2011, 2269, 2554, 2872, 3225, 3619, 4054, 4540, 5077, 5671, 6332

Offset: 0

Gus Wiseman, Nov 17 2023

nonn

			The partition y = (4,2,1) has semi-sums {3,5,6} which are missing 4, so y is counted under a(7).
The a(7) = 1 through a(13) = 10 partitions:
  (4,2,1)  (4,3,1)  (5,3,1)  (5,3,2)  (5,4,2)  (6,4,2)    (6,4,3)
           (5,2,1)  (6,2,1)  (5,4,1)  (6,3,2)  (6,5,1)    (6,5,2)
                             (6,3,1)  (6,4,1)  (7,3,2)    (7,4,2)
                             (7,2,1)  (7,3,1)  (7,4,1)    (7,5,1)
                                      (8,2,1)  (8,3,1)    (8,3,2)
                                               (9,2,1)    (8,4,1)
                                               (5,4,2,1)  (9,3,1)
                                               (6,3,2,1)  (10,2,1)
                                                          (6,4,2,1)
                                                          (7,3,2,1)

For parts instead of sums we have A238007:
- complement A001227
- non-strict complement A034296, ranks A073491
- non-strict  A239955, ranks A073492
The non-strict version is A367403.
The non-strict complement is A367402.
The complement is counted by A367410.
The non-binary version is A365831:
- non-strict complement A126796, ranks A325781
- complement A188431
- non-strict A365924, ranks A365830
A000009 counts partitions covering an initial interval, ranks A055932.
A046663 counts partitions w/o submultiset summing to k, strict A365663.
A365543 counts partitions w/ submultiset summing to k, strict A365661.
Cf. A000041, A002033, A261036, A264401, A276024, A284640, A304792, A364272.

Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&(d=Total/@Subsets[#, {2}];If[d=={},{}, Range[Min@@d,Max@@d]]!=Union[d])&]], {n,0,30}]

cp's OEIS Frontend

A367395 Number of strict integer partitions of n whose length is the sum of two distinct parts.

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A367399 Number of strict integer partitions of n whose length is not the sum of any two distinct parts.

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A367402 Number of integer partitions of n whose semi-sums cover an interval of positive integers.

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A367403 Number of integer partitions of n whose semi-sums do not cover an interval of positive integers.

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A367411 Number of strict integer partitions of n whose semi-sums do not cover an interval of positive integers.

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