A240861 -id:A240861 - cp's OEIS frontend

A367218 Number of integer partitions of n whose length can be written as a nonnegative linear combination of the distinct parts.

1, 1, 1, 2, 4, 6, 8, 13, 18, 26, 35, 50, 66, 92, 119, 160, 208, 275, 350, 457, 579, 742, 933, 1185, 1476, 1859, 2300, 2868, 3531, 4371, 5343, 6575, 8003, 9776, 11842, 14394, 17351, 20987, 25191, 30315, 36257, 43448, 51753, 61776, 73342, 87192, 103184, 122253, 144211

Offset: 0

Gus Wiseman, Nov 14 2023

nonn

The Heinz numbers of these partitions are given by A367226.

			The partition (4,2,1) has 3 = (2)+(1) or 3 = (1+1+1) so is counted under a(7).
The a(1) = 1 through a(7) = 13 partitions:
  (1)  (11)  (21)   (22)    (32)     (42)      (52)
             (111)  (31)    (41)     (51)      (61)
                    (211)   (221)    (321)     (322)
                    (1111)  (311)    (411)     (331)
                            (2111)   (2211)    (421)
                            (11111)  (3111)    (511)
                                     (21111)   (2221)
                                     (111111)  (3211)
                                               (4111)
                                               (22111)
                                               (31111)
                                               (211111)
                                               (1111111)

The following sequences count and rank integer partitions and finite sets according to whether their length is a subset-sum or linear combination of the parts. The current sequence is starred.
               sum-full   sum-free   comb-full  comb-free
              -------------------------------------------
  partitions:  A367212    A367213    A367218*   A367219
  strict:      A367214    A367215    A367220    A367221
  subsets:     A367216    A367217    A367222    A367223
  ranks:       A367224    A367225    A367226    A367227
A000041 counts integer partitions, strict A000009.
A002865 counts partitions whose length is a part, complement A229816.
A008284 counts partitions by length, strict A008289.
A240855 counts strict partitions whose length is a part, complement A240861.
A365046 counts combination-full subsets, differences of A364914.
Cf. A088314, A363225, A364350, A365073, A365311, A365918.

combs[n_,y_]:=With[{s=Table[{k,i},{k,y}, {i,0,Floor[n/k]}]}, Select[Tuples[s], Total[Times@@@#]==n&]];
Table[Length[Select[IntegerPartitions[n], combs[Length[#], Union[#]]!={}&]], {n,0,15}]

a(31)-a(48) from Chai Wah Wu, Nov 15 2023

A367220 Number of strict integer partitions of n whose length (number of parts) can be written as a nonnegative linear combination of the parts.

Original entry on oeis.org

1, 1, 0, 1, 1, 2, 3, 3, 4, 5, 7, 7, 10, 11, 15, 17, 22, 25, 32, 37, 46, 53, 65, 75, 90, 105, 124, 143, 168, 193, 224, 258, 297, 340, 390, 446, 509, 580, 660, 751, 852, 967, 1095, 1240, 1401, 1584, 1786, 2015, 2269, 2554, 2869, 3226, 3617, 4056, 4541, 5084

Offset: 0

Gus Wiseman, Nov 14 2023

nonn

The non-strict version is A367218.

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

The following sequences count and rank integer partitions and finite sets according to whether their length is a subset-sum or linear combination of the parts. The current sequence is starred.
               sum-full   sum-free   comb-full  comb-free
              -------------------------------------------
  partitions:  A367212    A367213    A367218    A367219
  strict:      A367214    A367215    A367220*   A367221
  subsets:     A367216    A367217    A367222    A367223
  ranks:       A367224    A367225    A367226    A367227
A000041 counts integer partitions, strict A000009.
A002865 counts partitions whose length is a part, complement A229816.
A188431 counts complete strict partitions, incomplete A365831.
A240855 counts strict partitions whose length is a part, complement A240861.
A364272 counts sum-full strict partitions, sum-free A364349.
A365046 counts combination-full subsets, differences of A364914.
Cf. A008289, A088314, A116861, A124506, A363225, A364346, A364350, A364916, A365073, A365311, A365312.

combs[n_,y_]:=With[{s=Table[{k,i},{k,y}, {i,0,Floor[n/k]}]}, Select[Tuples[s], Total[Times@@@#]==n&]];
Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&combs[Length[#], Union[#]]!={}&]], {n,0,15}]

A367221 Number of strict integer partitions of n whose length (number of parts) cannot be written as a nonnegative linear combination of the parts.

Original entry on oeis.org

0, 0, 1, 1, 1, 1, 1, 2, 2, 3, 3, 5, 5, 7, 7, 10, 10, 13, 14, 17, 18, 23, 24, 29, 32, 37, 41, 49, 54, 63, 72, 82, 93, 108, 122, 139, 159, 180, 204, 231, 261, 293, 331, 370, 415, 464, 518, 575, 641, 710, 789, 871, 965, 1064, 1177, 1294, 1428, 1569, 1729, 1897

Offset: 0

Gus Wiseman, Nov 14 2023

nonn

The non-strict version is A367219.

			The a(2) = 1 through a(16) = 10 strict partitions (A..G = 10..16):
  2  3  4  5  6  7   8   9   A   B    C    D    E    F    G
                 43  53  54  64  65   75   76   86   87   97
                         63  73  74   84   85   95   96   A6
                                 83   93   94   A4   A5   B5
                                 542  642  A3   B3   B4   C4
                                           652  752  C3   D3
                                           742  842  654  754
                                                     762  862
                                                     852  952
                                                     942  A42

The following sequences count and rank integer partitions and finite sets according to whether their length is a subset-sum or linear combination of the parts. The current sequence is starred.
               sum-full   sum-free   comb-full  comb-free
              -------------------------------------------
  partitions:  A367212    A367213    A367218    A367219
  strict:      A367214    A367215    A367220    A367221*
  subsets:     A367216    A367217    A367222    A367223
  ranks:       A367224    A367225    A367226    A367227
A000041 counts integer partitions, strict A000009.
A002865 counts partitions whose length is a part, complement A229816.
A124506 appears to count combination-free subsets, differences of A326083.
A188431 counts complete strict partitions, incomplete A365831.
A240855 counts strict partitions whose length is a part, complement A240861.
A364272 counts sum-full strict partitions, sum-free A364349.
Triangles:
A008284 counts partitions by length, strict A008289.
A046663 counts partitions of n without a subset-sum k, strict A365663.
A365541 counts subsets containing two distinct elements summing to k.
A365658 counts partitions by number of subset-sums, strict A365832.
Cf. A088314, A103580, A116861, A364346, A364350, A364533, A365312, A365380.

combs[n_,y_]:=With[{s=Table[{k,i},{k,y}, {i,0,Floor[n/k]}]}, Select[Tuples[s], Total[Times@@@#]==n&]];
Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&combs[Length[#], Union[#]]=={}&]], {n,0,30}]

A363260 Number of integer partitions of n with parts disjoint from first differences of parts, meaning no part is the difference of two consecutive parts.

Original entry on oeis.org

1, 1, 2, 2, 4, 5, 7, 10, 13, 17, 21, 28, 35, 46, 57, 70, 87, 110, 130, 165, 198, 238, 285, 349, 410, 498, 583, 702, 819, 983, 1136, 1353, 1570, 1852, 2137, 2520, 2898, 3390, 3891, 4540, 5191, 6028, 6889, 7951, 9082, 10450, 11884, 13650, 15508, 17728, 20113

Offset: 0

Gus Wiseman, Jul 19 2023

nonn

			The a(1) = 1 through a(8) = 13 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (111)  (22)    (32)     (33)      (43)       (44)
                    (31)    (41)     (51)      (52)       (53)
                    (1111)  (311)    (222)     (61)       (62)
                            (11111)  (411)     (322)      (71)
                                     (3111)    (331)      (332)
                                     (111111)  (511)      (611)
                                               (4111)     (2222)
                                               (31111)    (3311)
                                               (1111111)  (5111)
                                                          (41111)
                                                          (311111)
                                                          (11111111)

For length instead of differences we have A229816, strict A240861.
For all differences of pairs parts we have A364345.
For subsets of {1..n} instead of partitions we have A364463.
The strict case is A364464.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A323092 counts double-free partitions, ranks A320340.
A325325 counts partitions with distinct first-differences.
Cf. A002865, A025065, A108917, A236912, A237113, A237667, A320347, A326083, A363225, A364347.

Table[Length[Select[IntegerPartitions[n],Intersection[#,-Differences[#]]=={}&]],{n,0,30}]

from collections import Counter
from sympy.utilities.iterables import partitions
def A363260(n): return sum(1 for s,p in map(lambda x: (x[0],tuple(sorted(Counter(x[1]).elements()))), partitions(n,size=True)) if set(p).isdisjoint({p[i+1]-p[i] for i in range(s-1)})) # Chai Wah Wu, Sep 26 2023

A364533 Number of strict integer partitions of n containing the sum of no pair of distinct parts. A variation of sum-free strict partitions.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 3, 5, 5, 8, 7, 11, 11, 15, 15, 21, 22, 28, 32, 38, 40, 51, 55, 65, 74, 83, 94, 111, 119, 136, 160, 174, 196, 222, 252, 273, 315, 341, 391, 425, 477, 518, 602, 636, 719, 782, 886, 944, 1073, 1140, 1302, 1380, 1553, 1651, 1888, 1995, 2224, 2370

Offset: 0

Gus Wiseman, Aug 02 2023

nonn

			The a(1) = 1 through a(12) = 11 partitions (A..C = 10..12):
  1   2   3    4    5    6    7     8     9     A     B     C
          21   31   32   42   43    53    54    64    65    75
                    41   51   52    62    63    73    74    84
                              61    71    72    82    83    93
                              421   521   81    91    92    A2
                                          432   631   A1    B1
                                          531   721   542   543
                                          621         632   732
                                                      641   741
                                                      731   831
                                                      821   921

For subsets of {1..n} we have A085489, complement A088809.
The non-strict version is A236912, complement A237113, ranked by A364461.
Allowing re-used parts gives A364346.
The non-binary version is A364349, non-strict A237667 (complement A237668).
The linear combination-free version is A364350.
The complement in strict partitions is A364670, w/ re-used parts A363226.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A108917 counts knapsack partitions, strict A275972.
A151897 counts sum-free subsets, complement A364534.
A323092 counts double-free partitions, ranks A320340.
Cf. A007865, A025065, A240861, A325862, A326083, A364272, A364345, A364347.

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

A364463 Number of subsets of {1..n} with elements disjoint from first differences of elements.

Original entry on oeis.org

1, 2, 3, 6, 10, 18, 30, 54, 92, 167, 290, 525, 935, 1704, 3082, 5664, 10386, 19249, 35701, 66702, 124855, 234969, 443174, 839254, 1592925, 3032757, 5786153, 11066413, 21204855, 40712426, 78294085, 150815154, 290922900, 561968268, 1086879052, 2104570243

Offset: 0

Gus Wiseman, Jul 27 2023

nonn

In other words, no element is the difference of two consecutive elements.
From David A. Corneth, Aug 02 2023: (Start)
As subsets counted in a(n) are also counted in a(n+1) and {n+1} is a subset counted in a(n+1) but not a(n), a(n + 1) > a(n) for n >= 1.
As every subset counted in a(n + 1) that contains n+1 can be found from some subset counted in a(n) by appending n+1 and every subset counted in a(n) not containing n + 1 is counted in a(n + 1), a(n+1) <= 2*a(n). (End)

			The a(0) = 1 through a(5) = 18 subsets:
  {}  {}   {}   {}     {}       {}
      {1}  {1}  {1}    {1}      {1}
           {2}  {2}    {2}      {2}
                {3}    {3}      {3}
                {1,3}  {4}      {4}
                {2,3}  {1,3}    {5}
                       {1,4}    {1,3}
                       {2,3}    {1,4}
                       {3,4}    {1,5}
                       {2,3,4}  {2,3}
                                {2,5}
                                {3,4}
                                {3,5}
                                {4,5}
                                {1,3,5}
                                {2,3,4}
                                {3,4,5}
                                {2,3,4,5}

For all differences of pairs of elements we have A007865.
For partitions instead of subsets we have A363260, strict A364464.
The complement is counted by A364466.
A000041 counts integer partitions, strict A000009.
A364465 counts subsets with distinct first differences, partitions A325325.
Cf. A011782, A025065, A229816, A236912, A237113, A237667, A240861, A320347, A323092, A326083, A364347.

Table[Length[Select[Subsets[Range[n]],Intersection[#,Differences[#]]=={}&]],{n,0,10}]

from itertools import combinations
def A364463(n): return sum(1 for l in range(n+1) for c in combinations(range(1,n+1),l) if set(c).isdisjoint({c[i+1]-c[i] for i in range(l-1)})) # Chai Wah Wu, Sep 26 2023

a(n) < a(n + 1) <= 2 * a(n). - David A. Corneth, Aug 02 2023

a(21)-a(29) from David A. Corneth, Aug 02 2023
a(30)-a(32) from Chai Wah Wu, Sep 26 2023
a(33)-a(35) from Chai Wah Wu, Sep 27 2023

A364464 Number of strict integer partitions of n where no part is the difference of two consecutive parts.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 2, 4, 4, 6, 5, 8, 9, 12, 13, 16, 16, 21, 23, 29, 34, 38, 41, 49, 57, 64, 73, 86, 95, 110, 120, 135, 160, 171, 197, 219, 247, 277, 312, 342, 386, 431, 476, 527, 598, 640, 727, 796, 893, 966, 1097, 1178, 1327, 1435, 1602, 1740, 1945, 2084, 2337

Offset: 0

Gus Wiseman, Jul 30 2023

nonn

In other words, the parts are disjoint from the first differences.

			The strict partition y = (9,5,3,1) has differences (4,2,2), and these are disjoint from the parts, so y is counted under a(18).
The a(1) = 1 through a(9) = 6 strict partitions:
  (1)  (2)  (3)  (4)    (5)    (6)    (7)    (8)    (9)
                 (3,1)  (3,2)  (5,1)  (4,3)  (5,3)  (5,4)
                        (4,1)         (5,2)  (6,2)  (7,2)
                                      (6,1)  (7,1)  (8,1)
                                                    (4,3,2)
                                                    (5,3,1)

For length instead of differences we have A240861, non-strict A229816.
For all differences of pairs of elements we have A364346, for subsets A007865.
For subsets instead of strict partitions we have A364463, complement A364466.
The non-strict version is A363260.
The complement is counted by A364536, non-strict A364467.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A120641 counts strict double-free partitions, non-strict A323092.
A320347 counts strict partitions w/ distinct differences, non-strict A325325.
Cf. A002865, A025065, A236912, A237667, A275972, A363226, A364345, A364465.

Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Intersection[#,-Differences[#]]=={}&]],{n,0,15}]

from collections import Counter
from sympy.utilities.iterables import partitions
def A364464(n): return sum(1 for s,p in map(lambda x: (x[0],tuple(sorted(Counter(x[1]).elements()))), filter(lambda p:max(p[1].values(),default=1)==1,partitions(n,size=True))) if set(p).isdisjoint({p[i+1]-p[i] for i in range(s-1)})) # Chai Wah Wu, Sep 26 2023

A364670 Number of strict integer partitions of n with a part equal to the sum of two distinct others. A variation of sum-full strict partitions.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 3, 1, 4, 3, 7, 6, 10, 10, 14, 16, 24, 25, 34, 39, 48, 59, 71, 81, 103, 120, 136, 166, 194, 226, 260, 312, 353, 419, 473, 557, 636, 742, 824, 974, 1097, 1266, 1418, 1646, 1837, 2124, 2356, 2717, 3029, 3469, 3830, 4383, 4884, 5547

Offset: 0

Gus Wiseman, Aug 03 2023

nonn

			The a(6) = 1 through a(16) = 10 strict partitions (A = 10):
  321  .  431  .  532   5321  642   5431  743   6432   853
                  541         651   6421  752   6531   862
                  4321        5421  7321  761   7431   871
                              6321        5432  7521   6532
                                          6431  9321   6541
                                          6521  54321  7432
                                          8321         7621
                                                       8431
                                                       A321
                                                       64321

For subsets of {1..n} we have A088809, complement A085489.
The non-strict version is A237113, complement A236912.
The non-binary complement is A237667, ranks A364532.
Allowing re-used parts gives A363226, non-strict A363225.
The non-binary version is A364272, non-strict A237668.
The complement is A364533, non-binary A364349.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A108917 counts knapsack partitions, strict A275972, ranks A299702.
A323092 counts double-free partitions, ranks A320340.
Cf. A007865, A025065, A093971, A111133, A151897, A240861, A325862, A364346, A364350, A364534.

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

A240855 Number of partitions p of n into distinct parts including the number of parts.

Original entry on oeis.org

0, 1, 0, 1, 0, 1, 2, 1, 2, 3, 4, 3, 5, 6, 8, 9, 10, 12, 16, 18, 22, 25, 29, 34, 41, 48, 55, 64, 74, 84, 98, 114, 130, 150, 170, 195, 222, 252, 287, 328, 371, 420, 475, 536, 604, 682, 766, 862, 970, 1088, 1218, 1365, 1526, 1704, 1904, 2124, 2366, 2637, 2934

Offset: 0

Clark Kimberling, Apr 14 2014

nonn

			a(10) counts these 4 partitions:  82, 631, 532, 4321.

Alois P. Heinz, Table of n, a(n) for n = 0..4000 (first 101 terms from John Tyler Rascoe)
Atul Dixit, Gaurav Kumar, and Aviral Srivastava, Non-Rascoe partitions and a rank parity function associated to the Rogers-Ramanujan partitions, arXiv:2508.04359 [math.CO], 2025. See references.

Cf. A000009, A008967, A240861.

h:= (p, i)-> add(coeff(p, x, j)*x^j, j=i+1..degree(p)):
b:= proc(n, i, p) option remember; `if`(i*(i+1)/2 [h(g[1], i), g[2]])(b(n, i-1, p)+
     (f-> f+[0, coeff(f[1], x, i)])(b(n-i, min(n-i, i-1), p+1)))))
    end:
a:= n-> b(n$2, 0)[2]:
seq(a(n), n=0..58);  # Alois P. Heinz, Mar 14 2024

z = 40;
f[n_] := f[n] = Select[IntegerPartitions[n], Max[Length /@ Split@#] == 1 &];
Table[Count[f[n], p_ /; MemberQ[p, Length[p]]], {n, 0, z}] (* this sequence *)
Table[Count[f[n], p_ /; !MemberQ[p, Length[p]]], {n, 0, z}] (* A240861 *)

p_q(k) = {prod(j=1,k, 1-q^j);}
mGB_q(N,M) = {p_q(N+M)/(p_q(M)*(p_q(N)^2));}
A_q(N) = {my(q='q+O('q^N), g=sum(i=1, N, sum(j=1, i, q^((i*(i+1)/2)+(j*(j-1))) * mGB_q(j-1,i-j))));
concat([0], Vec(g)) }
A_q(50) \\ John Tyler Rascoe, Mar 13 2024

a(n) = A000009(n) - A240861(n).

G.f.: Sum_{i>0} Sum_{j=1..i} q^((i*(i+1)/2) + j*(j-1)) * [j-1,i-j]q, where [N,M]_q = Product{j=1..N+M}(1-q^j) / (Product_{j=1..M}(1-q^j) * (Product_{j=1..N}(1-q^j))^2). - John Tyler Rascoe, Mar 13 2024

cp's OEIS Frontend

A367218 Number of integer partitions of n whose length can be written as a nonnegative linear combination of the distinct parts.

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A367220 Number of strict integer partitions of n whose length (number of parts) can be written as a nonnegative linear combination of the parts.

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A367221 Number of strict integer partitions of n whose length (number of parts) cannot be written as a nonnegative linear combination of the parts.

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A363260 Number of integer partitions of n with parts disjoint from first differences of parts, meaning no part is the difference of two consecutive parts.

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A364533 Number of strict integer partitions of n containing the sum of no pair of distinct parts. A variation of sum-free strict partitions.

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A364463 Number of subsets of {1..n} with elements disjoint from first differences of elements.

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A364464 Number of strict integer partitions of n where no part is the difference of two consecutive parts.

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A364670 Number of strict integer partitions of n with a part equal to the sum of two distinct others. A variation of sum-full strict partitions.

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A240855 Number of partitions p of n into distinct parts including the number of parts.

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