cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A365378 Number of integer partitions with sum < n whose distinct parts cannot be linearly combined using nonnegative coefficients to obtain n.

Original entry on oeis.org

0, 0, 0, 1, 1, 4, 2, 9, 5, 13, 10, 28, 7, 45, 25, 51, 32, 101, 31, 148, 50, 166, 106, 291, 47, 374, 176, 450, 179, 721, 121, 963, 285, 1080, 474, 1534, 200, 2140, 712, 2407, 599, 3539, 481, 4546, 1014, 4885
Offset: 0

Views

Author

Gus Wiseman, Sep 04 2023

Keywords

Examples

			The partition (5,2,2) has distinct parts {2,5} and has 11 = 3*2 + 1*5, so is not counted under a(11).
The partition (4,2,2) cannot be linearly combined to obtain 9, so is counted under a(9).
The partition (4,2,2) has distinct parts {2,4} and has 10 = 5*2 + 0*4, so is not counted under a(10).
The a(3) = 1 through a(10) = 10 partitions:
  (2)  (3)  (2)   (4)  (2)    (3)   (2)     (3)
            (3)   (5)  (3)    (5)   (4)     (4)
            (4)        (4)    (6)   (5)     (6)
            (22)       (5)    (7)   (6)     (7)
                       (6)    (33)  (7)     (8)
                       (22)         (8)     (9)
                       (33)         (22)    (33)
                       (42)         (42)    (44)
                       (222)        (44)    (63)
                                    (62)    (333)
                                    (222)
                                    (422)
                                    (2222)

Crossrefs

The complement for subsets is A365073, positive coefficients A088314.
For strict partitions we have A365312, positive coefficients A088528.
For positive coefficients we have A365323.
The complement is counted by A365379.
The version for subsets is A365380, positive coefficients A365322.
The relatively prime case is A365382.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A116861 and A364916 count linear combinations of strict partitions.
A364350 counts combination-free strict partitions, non-strict A364915.
A364839 counts combination-full strict partitions, non-strict A364913.
Cf. A237668, A363225, A364272, A364345, A364914, A365320.

Programs

  • Mathematica
    combs[n_,y_]:=With[{s=Table[{k,i},{k,y},{i,0,Floor[n/k]}]},Select[Tuples[s],Total[Times@@@#]==n&]];
Table[Length[Select[Join@@IntegerPartitions/@Range[n-1],combs[n,Union[#]]=={}&]],{n,0,10}]
  • Python
    from sympy.utilities.iterables import partitions
def A365378(n):
    a = {tuple(sorted(set(p))) for p in partitions(n)}
    return sum(1 for m in range(1,n) for b in partitions(m) if not any(set(d).issubset(set(b)) for d in a)) # Chai Wah Wu, Sep 13 2023

Extensions

a(21)-a(45) from Chai Wah Wu, Sep 13 2023

A365379 Number of integer partitions with sum <= n whose distinct parts can be linearly combined using nonnegative coefficients to obtain n.

Original entry on oeis.org

0, 1, 3, 5, 10, 14, 27, 35, 61, 83, 128, 166, 264, 327, 482, 632, 882, 1110, 1565, 1938, 2663, 3339, 4401, 5471, 7290, 8921, 11555, 14291, 18280, 22303, 28507, 34507, 43534, 52882, 65798, 79621, 98932, 118629, 146072, 175562, 214708, 256351, 312583, 371779
Offset: 0

Views

Author

Gus Wiseman, Sep 04 2023

Keywords

Examples

			The partition (4,2,2) cannot be linearly combined to obtain 9, so is not counted under a(9). On the other hand, the same partition (4,2,2) has distinct parts {2,4} and has 10 = 1*2 + 2*4, so is counted under a(10).
The a(1) = 1 through a(5) = 14 partitions:
  (1)  (1)   (1)    (1)     (1)
       (2)   (3)    (2)     (5)
       (11)  (11)   (4)     (11)
             (21)   (11)    (21)
             (111)  (21)    (31)
                    (22)    (32)
                    (31)    (41)
                    (111)   (111)
                    (211)   (211)
                    (1111)  (221)
                            (311)
                            (1111)
                            (2111)
                            (11111)

Crossrefs

For subsets with positive coefficients we have A088314, complement A088528.
The case of strict partitions with positive coefficients is also A088314.
The version for subsets is A365073, complement A365380.
The case of strict partitions is A365311, complement A365312.
The complement is counted by A365378.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A116861 and A364916 count linear combinations of strict partitions.
A364350 counts combination-free strict partitions, non-strict A364915.
A364839 counts combination-full strict partitions, non-strict A364913.
Cf. A237668, A363225, A364272, A364345, A364914, A365320, A365382.

Programs

  • Mathematica
    combs[n_,y_]:=With[{s=Table[{k,i},{k,y},{i,0,Floor[n/k]}]},Select[Tuples[s],Total[Times@@@#]==n&]];
Table[Length[Select[Join@@Array[IntegerPartitions,n],combs[n,Union[#]]!={}&]],{n,0,10}]
  • Python
    from sympy.utilities.iterables import partitions
def A365379(n):
    a = {tuple(sorted(set(p))) for p in partitions(n)}
    return sum(1 for m in range(1,n+1) for b in partitions(m) if any(set(d).issubset(set(b)) for d in a)) # Chai Wah Wu, Sep 13 2023

Extensions

a(21)-a(43) from Chai Wah Wu, Sep 13 2023

A365323 Number of integer partitions with sum < n whose distinct parts cannot be linearly combined using all positive coefficients to obtain n.

Original entry on oeis.org

0, 0, 1, 1, 4, 3, 9, 7, 15, 16, 29, 23, 47, 43, 74, 65, 114, 100, 174, 153, 257, 228, 368, 312, 530, 454, 736, 645, 1025, 902, 1402, 1184, 1909, 1626, 2618, 2184, 3412, 2895, 4551, 3887, 5966, 5055, 7796, 6509, 10244, 8462, 13060, 10881, 16834, 14021, 21471
Offset: 1

Views

Author

Gus Wiseman, Sep 12 2023

Keywords

Examples

			The partition y = (3,3,2) has distinct parts {2,3}, and we have 9 = 3*2 + 1*3, so y is not counted under a(9).
The a(3) = 1 through a(10) = 16 partitions:
  (2)  (3)  (2)    (4)    (2)      (3)    (2)        (3)
            (3)    (5)    (3)      (5)    (4)        (4)
            (4)    (3,2)  (4)      (6)    (5)        (6)
            (2,2)         (5)      (7)    (6)        (7)
                          (6)      (3,3)  (7)        (8)
                          (2,2)    (4,3)  (8)        (9)
                          (3,3)    (5,2)  (2,2)      (3,3)
                          (4,2)           (4,2)      (4,4)
                          (2,2,2)         (4,3)      (5,2)
                                          (4,4)      (5,3)
                                          (5,3)      (5,4)
                                          (6,2)      (6,3)
                                          (2,2,2)    (7,2)
                                          (4,2,2)    (3,3,3)
                                          (2,2,2,2)  (4,3,2)
                                                     (5,2,2)

Links

Crossrefs

Complement for subsets: A088314 or A365042, nonnegative A365073 or A365542.
For strict partitions we have A088528, nonnegative coefficients A365312.
For length-2 subsets we have A365321 (we use n instead of n-1).
For subsets we have A365322 or A365045, nonnegative coefficients A365380.
For nonnegative coefficients we have A365378, complement A365379.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A116861 and A364916 count linear combinations of strict partitions.
A364350 counts combination-free strict partitions, non-strict A364915.
A364839 counts combination-full strict partitions, non-strict A364913.
Cf. A237668, A363225, A364272, A364345, A364914, A365320, A365382.

Programs

  • Mathematica
    combp[n_,y_]:=With[{s=Table[{k,i},{k,y},{i,1,Floor[n/k]}]},Select[Tuples[s],Total[Times@@@#]==n&]];
Table[Length[Select[Join@@IntegerPartitions/@Range[n-1],combp[n,Union[#]]=={}&]],{n,10}]
  • Python
    from sympy.utilities.iterables import partitions
def A365323(n):
    a = {tuple(sorted(set(p))) for p in partitions(n)}
    return sum(1 for k in range(1,n) for d in partitions(k) if tuple(sorted(set(d))) not in a) # Chai Wah Wu, Sep 12 2023

Extensions

a(21)-a(51) from Chai Wah Wu, Sep 12 2023

A365383 Triangle read by rows where T(n,k) is the number of integer partitions of n that can be linearly combined with nonnegative coefficients to obtain k.

Original entry on oeis.org

1, 2, 1, 3, 2, 2, 5, 3, 4, 3, 7, 5, 6, 6, 6, 11, 7, 9, 8, 9, 7, 15, 11, 13, 13, 14, 13, 14, 22, 15, 19, 17, 20, 17, 20, 16, 30, 22, 26, 26, 27, 26, 28, 26, 27, 42, 30, 37, 34, 39, 33, 40, 34, 39, 34, 56, 42, 50, 49, 52, 50, 54, 51, 54, 53, 53
Offset: 0

Views

Author

Gus Wiseman, Sep 08 2023

Keywords

Comments

Conjecture: The rows eventually become periodic with period n if extended further. For example, row n = 8 begins:
22, 15, 19, 17, 20, 17, 20, 16,
22, 17, 20, 17, 21, 17, 20, 17,
22, 17, 20, 17, 21, 17, 20, 17, ...

Examples

			Triangle begins:
   1
   2   1
   3   2   2
   5   3   4   3
   7   5   6   6   6
  11   7   9   8   9   7
  15  11  13  13  14  13  14
  22  15  19  17  20  17  20  16
  30  22  26  26  27  26  28  26  27
  42  30  37  34  39  33  40  34  39  34
  56  42  50  49  52  50  54  51  54  53  53
  77  56  68  64  71  63  73  63  71  65  70  62
 101  77  91  89  95  90  97  93  97  97  98  94  99
 135 101 122 115 127 115 130 114 131 119 130 117 132 116
 176 135 159 156 165 157 170 161 167 168 166 165 172 164 166
Row n = 6 counts the following partitions:
  (6)       (51)      (51)      (51)      (51)      (51)
  (51)      (411)     (42)      (411)     (42)      (411)
  (42)      (321)     (411)     (33)      (411)     (321)
  (411)     (3111)    (321)     (321)     (321)     (3111)
  (33)      (2211)    (3111)    (3111)    (3111)    (2211)
  (321)     (21111)   (222)     (2211)    (222)     (21111)
  (3111)    (111111)  (2211)    (21111)   (2211)    (111111)
  (222)               (21111)   (111111)  (21111)
  (2211)              (111111)            (111111)
  (21111)
  (111111)

Crossrefs

Column k = 0 is A000041, strict A000009.
The version for subsets is A365381, main diagonal A365376.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A116861 and A364916 count linear combinations of strict partitions.
A364350 counts combination-free strict partitions, non-strict A364915.
A364839 counts combination-full strict partitions, non-strict A364913.
Cf. A088314, A088528, A237668, A363225, A364345, A365073, A365311, A365320, A365378, A365382.

Programs

  • Mathematica
    combu[n_,y_]:=With[{s=Table[{k,i},{k,Union[y]},{i,0,Floor[n/k]}]},Select[Tuples[s],Total[Times@@@#]==n&]];
Table[Length[Select[IntegerPartitions[n],combu[k,#]!={}&]],{n,0,12},{k,0,n-1}]
Showing 1-4 of 4 results.

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