A364349 -id:A364349 - cp's OEIS frontend

A365663 Triangle read by rows where T(n,k) is the number of strict integer partitions of n without a subset summing to k.

1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 2, 2, 3, 2, 2, 3, 3, 3, 3, 3, 3, 3, 4, 3, 5, 3, 4, 3, 5, 5, 4, 5, 5, 4, 5, 5, 5, 6, 5, 6, 7, 6, 5, 6, 5, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 9, 8, 8, 8, 11, 8, 8, 8, 9, 8, 10, 11, 10, 10, 10, 10, 10, 10, 10, 10, 11, 10, 12, 13, 11, 13, 11, 12, 15, 12, 11, 13, 11, 13, 12

Offset: 2

Gus Wiseman, Sep 17 2023

Warning: Do not confuse with the non-strict version A046663.

Rows are palindromes.

			Triangle begins:
  1
  1  1
  1  2  1
  2  2  2  2
  2  2  3  2  2
  3  3  3  3  3  3
  3  4  3  5  3  4  3
  5  5  4  5  5  4  5  5
  5  6  5  6  7  6  5  6  5
  7  7  7  7  7  7  7  7  7  7
  8  9  8  8  8 11  8  8  8  9  8
Row n = 8 counts the following strict partitions:
  (8)    (8)      (8)    (8)      (8)    (8)      (8)
  (6,2)  (7,1)    (7,1)  (7,1)    (7,1)  (7,1)    (6,2)
  (5,3)  (5,3)    (6,2)  (6,2)    (6,2)  (5,3)    (5,3)
         (4,3,1)         (5,3)           (4,3,1)
                         (5,2,1)

Robert Price, Table of n, a(n) for n = 2..1226
P. Erdős, J. L. Nicolas and A. Sárközy, On the number of partitions of n without a given subsum (I), Discrete Math., 75 (1989), 155-166 = Annals Discrete Math. Vol. 43, Graph Theory and Combinatorics 1988, ed. B. Bollobas.

Columns k = 0 and k = n are A025147.
The non-strict version is A046663, central column A006827.
Central column n = 2k is A321142.
The complement for subsets instead of strict partitions is A365381.
The complement is A365661, non-strict A365543, central column A237258.
Row sums are A365922.
A000009 counts subsets summing to n.
A000124 counts distinct possible sums of subsets of {1..n}.
A124506 appears to count combination-free subsets, differences of A326083.
A364272 counts sum-full strict partitions, sum-free A364349.
A364350 counts combination-free strict partitions, complement A364839.
Cf. A002219, A108796, A108917, A122768, A275972, A299701, A304792, A364916, A365311, A365376, A365541.

Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&FreeQ[Total/@Subsets[#],k]&]], {n,2,15},{k,1,n-1}]

A124506 Number of numerical semigroups with Frobenius number n; that is, numerical semigroups for which the largest integer not belonging to them is n.

Original entry on oeis.org

1, 1, 2, 2, 5, 4, 11, 10, 21, 22, 51, 40, 106, 103, 200, 205, 465, 405, 961, 900, 1828, 1913, 4096, 3578, 8273, 8175, 16132, 16267, 34903, 31822, 70854, 68681, 137391, 140661, 292081, 270258, 591443, 582453, 1156012

Offset: 1

P. A. Garcia-Sanchez (pedro(AT)ugr.es), Dec 18 2006

From Gus Wiseman, Aug 28 2023: (Start)
Appears to be the number of subsets of {1..n} containing n such that no element can be written as a nonnegative linear combination of the others, first differences of A326083. For example, the a(1) = 1 through a(8) = 10 subsets are:
  {1}  {2}  {3}    {4}    {5}      {6}      {7}        {8}
            {2,3}  {3,4}  {2,5}    {4,6}    {2,7}      {3,8}
                          {3,5}    {5,6}    {3,7}      {5,8}
                          {4,5}    {4,5,6}  {4,7}      {6,8}
                          {3,4,5}           {5,7}      {7,8}
                                            {6,7}      {3,7,8}
                                            {3,5,7}    {5,6,8}
                                            {4,5,7}    {5,7,8}
                                            {4,6,7}    {6,7,8}
                                            {5,6,7}    {5,6,7,8}
                                            {4,5,6,7}
Note that these subsets do not all generate numerical semigroups, as their GCD is unrestricted, cf. A358392. The complement is counted by A365046, first differences of A364914.
(End)

			a(1) = 1 via <2,3> = {0,2,3,4,...}; the largest missing number is 1.
a(2) = 1 via <3,4,5> = {0,3,4,5,...}; the largest missing number is 2.
a(3) = 2 via <2,5> = {0,2,4,5,...}; and <4,5,6,7> = {0,4,5,6,7,...} where in both the largest missing number is 3.
a(4) = 2 via <3,5,7> = {0,3,5,6,7,...} and <5,6,7,8,9> = {5,6,7,8,9,...} where in both the largest missing number is 4.

S. R. Finch, Monoids of natural numbers
Manuel Delgado, Neeraj Kumar, and Claude Marion, On counting numerical semigroups by maximum primitive and Wilf's conjecture, arXiv:2501.04417 [math.CO], 2025. See p. 22.
S. R. Finch, Monoids of natural numbers, March 17, 2009. [Cached copy, with permission of the author]
J. C. Rosales, P. A. Garcia-Sanchez, J. I. Garcia-Garcia, and J. A. Jimenez-Madrid, Fundamental gaps in numerical semigroups, Journal of Pure and Applied Algebra 189 (2004) 301-313.
Clayton Cristiano Silva, Irreducible Numerical Semigroups, University of Campinas, São Paulo, Brazil (2019).

Cf. A158206. [From Steven Finch, Mar 13 2009]
A288728 counts sum-free sets, first differences of A007865.
A364350 counts combination-free partitions, complement A364839.
Cf. A085489, A088809, A093971, A103580, A116861, A151897, A237668, A308546, A326020, A326083, A364349, A365069.

The sequence was originally generated by a C program and a Haskell script. The sequence can be obtained by using the function NumericalSemigroupsWithFrobeniusNumber included in the numericalsgps GAP package.

A364346 Number of strict integer partitions of n such that there is no ordered triple of parts (a,b,c) (repeats allowed) satisfying a + b = c. A variation of sum-free strict partitions.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 2, 4, 4, 5, 5, 8, 9, 11, 11, 16, 16, 20, 20, 25, 30, 34, 38, 42, 50, 58, 64, 73, 80, 90, 105, 114, 128, 148, 158, 180, 201, 220, 241, 277, 306, 333, 366, 404, 447, 497, 544, 592, 662, 708, 797, 861, 954, 1020, 1131, 1226, 1352, 1456, 1600

Offset: 0

Gus Wiseman, Jul 22 2023

nonn

			The a(1) = 1 through a(14) = 11 partitions (A..E = 10..14):
  1   2   3   4    5    6    7    8    9     A    B     C     D     E
              31   32   51   43   53   54    64   65    75    76    86
                   41        52   62   72    73   74    93    85    95
                             61   71   81    82   83    A2    94    A4
                                       531   91   92    B1    A3    B3
                                                  A1    543   B2    C2
                                                  641   732   C1    D1
                                                  731   741   652   851
                                                        831   751   932
                                                              832   941
                                                              931   A31

For subsets of {1..n} we have A007865 (sum-free sets), differences A288728.
For sums of any length > 1 we have A364349, non-strict A237667.
The complement is counted by A363226, non-strict A363225.
The non-strict version is A364345, ranks A364347, complement A364348.
A000041 counts partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A236912 counts sum-free partitions not re-using parts, complement A237113.
A323092 counts double-free partitions, ranks A320340.
Cf. A002865, A025065, A085489, A093971, A108917, A111133, A240861, A275972, A320347, A325862, A326083, A363260.

Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Select[Tuples[#,3],#[[1]]+#[[2]]==#[[3]]&]=={}&]],{n,0,15}]

from collections import Counter
from itertools import combinations_with_replacement
from sympy.utilities.iterables import partitions
def A364346(n): return sum(1 for p in partitions(n) if max(p.values(),default=1)==1 and not any(q[0]+q[1]==q[2] for q in combinations_with_replacement(sorted(Counter(p).elements()),3))) # Chai Wah Wu, Sep 20 2023

A367213 Number of integer partitions of n whose length (number of parts) is not equal to the sum of any submultiset.

Original entry on oeis.org

0, 0, 1, 1, 2, 2, 5, 4, 7, 8, 12, 13, 19, 21, 29, 33, 45, 49, 67, 73, 97, 108, 139, 152, 196, 217, 274, 303, 379, 420, 523, 579, 709, 786, 960, 1061, 1285, 1423, 1714, 1885, 2265, 2498, 2966, 3280, 3881, 4268, 5049, 5548, 6507, 7170, 8391, 9194, 10744, 11778, 13677

Offset: 0

Gus Wiseman, Nov 12 2023

nonn

These partitions are necessarily incomplete (A365924).

Are there any decreases after the initial terms?

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

Chai Wah Wu, Table of n, a(n) for n = 0..65

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 partitions, strict A000009.
A002865 counts partitions whose length is a part, complement A229816.
A007865/A085489/A151897 count certain types of sum-free subsets.
A108917 counts knapsack partitions, non-knapsack A366754.
A126796 counts complete partitions, incomplete A365924.
A237667 counts sum-free partitions, sum-full A237668.
A304792 counts subset-sums of partitions, strict A365925.
Triangles:
A008284 counts partitions by length, strict A008289.
A046663 counts partitions of n without a subset-sum k, strict A365663.
A365543 counts partitions of n with a subset-sum k, strict A365661.
A365658 counts partitions by number of subset-sums, strict A365832.
Cf. A000700, A124506, A238628, A240861, A364349, A364531, A365045, A365381, A365918, A366320.

Table[Length[Select[IntegerPartitions[n], FreeQ[Total/@Subsets[#], Length[#]]&]], {n,0,10}]

a(41)-a(54) from Chai Wah Wu, Nov 13 2023

A367214 Number of strict integer partitions of n whose length (number of parts) is equal to the sum of some submultiset.

Original entry on oeis.org

1, 1, 0, 1, 0, 1, 2, 2, 3, 4, 5, 5, 7, 8, 10, 12, 14, 17, 21, 25, 30, 36, 43, 51, 60, 71, 83, 97, 113, 132, 153, 178, 205, 238, 272, 315, 360, 413, 471, 539, 613, 698, 792, 899, 1018, 1153, 1302, 1470, 1658, 1867, 2100, 2362, 2652, 2974, 3335, 3734, 4178, 4672

Offset: 0

Gus Wiseman, Nov 12 2023

nonn

These partitions have Heinz numbers A367224 /\ A005117.

			The strict partition (6,4,3,2,1) has submultisets {1,4} and {2,3} with sum 5 so is counted under a(16).
The a(1) = 1 through a(10) = 5 strict partitions:
  (1)  .  (2,1)  .  (3,2)  (4,2)    (5,2)    (6,2)    (7,2)    (8,2)
                           (3,2,1)  (4,2,1)  (4,3,1)  (4,3,2)  (5,3,2)
                                             (5,2,1)  (5,3,1)  (6,3,1)
                                                      (6,2,1)  (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.
A088809/A093971/A364534 count certain types of sum-full subsets.
A188431 counts complete strict partitions, incomplete A365831.
A240855 counts strict partitions whose length is a part, complement A240861.
A275972 counts strict knapsack partitions, non-strict A108917.
A364272 counts sum-full strict partitions, sum-free A364349.
A365925 counts subset-sums of strict partitions, non-strict A304792.
Triangles:
A008289 counts strict partitions by length, non-strict A008284.
A365661 counts strict partitions with a subset-sum k, non-strict A365543.
A365832 counts strict partitions by subset-sums, non-strict A365658.
Cf. A002865, A126796, A237113, A237668, A238628, A363225, A364346, A364350, A364533, A365311, A365922.

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

A367215 Number of strict integer partitions of n whose length (number of parts) is not equal to the sum of any subset.

Original entry on oeis.org

0, 0, 1, 1, 2, 2, 2, 3, 3, 4, 5, 7, 8, 10, 12, 15, 18, 21, 25, 29, 34, 40, 46, 53, 62, 71, 82, 95, 109, 124, 143, 162, 185, 210, 240, 270, 308, 347, 393, 443, 500, 562, 634, 711, 798, 895, 1002, 1120, 1252, 1397, 1558, 1735, 1930, 2146, 2383, 2644, 2930, 3245

Offset: 0

Gus Wiseman, Nov 12 2023

nonn

These partitions have Heinz numbers A367225 /\ A005117.

			The a(2) = 1 through a(11) = 7 strict partitions:
  (2)  (3)  (4)    (5)    (6)    (7)    (8)    (9)    (10)     (11)
            (3,1)  (4,1)  (5,1)  (4,3)  (5,3)  (5,4)  (6,4)    (6,5)
                                 (6,1)  (7,1)  (6,3)  (7,3)    (7,4)
                                               (8,1)  (9,1)    (8,3)
                                                      (5,4,1)  (10,1)
                                                               (5,4,2)
                                                               (6,4,1)
The a(2) = 1 through a(15) = 15 strict partitions (A..F = 10..15):
  2  3  4   5   6   7   8   9   A    B    C    D    E     F
        31  41  51  43  53  54  64   65   75   76   86    87
                    61  71  63  73   74   84   85   95    96
                            81  91   83   93   94   A4    A5
                                541  A1   B1   A3   B3    B4
                                     542  642  C1   D1    C3
                                     641  651  652  752   E1
                                          741  742  761   654
                                               751  842   762
                                               841  851   852
                                                    941   861
                                                    6521  942
                                                          951
                                                          A41
                                                          7521

Chai Wah Wu, Table of n, a(n) for n = 0..118

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.
A007865/A085489/A151897 count certain types of sum-free subsets.
A124506 appears to count combination-free subsets, differences of A326083.
A188431 counts complete strict partitions, incomplete A365831.
A237667 counts sum-free partitions, ranks A364531.
A240861 counts strict partitions with length not a part, complement A240855.
A275972 counts strict knapsack partitions, non-strict A108917.
A364349 counts sum-free strict partitions, sum-full A364272.
Triangles:
A008289 counts strict partitions by length, non-strict A008284.
A365661 counts strict partitions with a subset-sum k, non-strict A365543.
A365663 counts strict partitions without a subset-sum k, non-strict A046663.
A365832 counts strict partitions by subset-sums, non-strict A365658.
Cf. A002865, A229816, A238628, A364346, A364350, A365312, A365922, A366320.

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

A367216 Number of subsets of {1..n} whose cardinality is equal to the sum of some subset.

Original entry on oeis.org

1, 2, 3, 5, 10, 20, 40, 82, 169, 348, 716, 1471, 3016, 6171, 12605, 25710, 52370, 106539, 216470, 439310, 890550, 1803415, 3648557, 7375141, 14896184, 30065129, 60639954, 122231740, 246239551, 495790161, 997747182, 2006969629, 4035274292, 8110185100, 16293958314, 32724456982

Offset: 0

Gus Wiseman, Nov 12 2023

nonn

			The a(0) = 1 through a(4) = 10 subsets:
  {}  {}   {}     {}       {}
      {1}  {1}    {1}      {1}
           {1,2}  {1,2}    {1,2}
                  {2,3}    {2,3}
                  {1,2,3}  {2,4}
                           {1,2,3}
                           {1,2,4}
                           {1,3,4}
                           {2,3,4}
                           {1,2,3,4}

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
A000009 counts subsets summing to n.
A000124 counts distinct possible sums of subsets of {1..n}.
A002865 counts partitions whose length is a part, complement A229816.
A007865/A085489/A151897 count certain types of sum-free subsets.
A088809/A093971/A364534 count certain types of sum-full subsets.
A237668 counts sum-full partitions, ranks A364532.
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.
Triangles:
A365381 counts sets with a subset summing to k, without A366320.
A365541 counts sets containing two distinct elements summing to k.
Cf. A068911, A095944, A103580, A288728, A326080, A326083, A365376, A365544.

Table[Length[Select[Subsets[Range[n]], MemberQ[Total/@Subsets[#], Length[#]]&]], {n,0,10}]

a(n) = 2^n - A367217(n). - Chai Wah Wu, Nov 14 2023

a(16)-a(28) from Chai Wah Wu, Nov 14 2023

a(29)-a(35) from Max Alekseyev, Feb 25 2025

A367217 Number of subsets of {1..n} whose cardinality is not equal to the sum of any subset.

Original entry on oeis.org

0, 0, 1, 3, 6, 12, 24, 46, 87, 164, 308, 577, 1080, 2021, 3779, 7058, 13166, 24533, 45674, 84978, 158026, 293737, 545747, 1013467, 1881032, 3489303, 6468910, 11985988, 22195905, 41080751, 75994642, 140514019, 259693004, 479749492, 885910870, 1635281386

Offset: 0

Gus Wiseman, Nov 12 2023

nonn

			The a(2) = 1 through a(5) = 12 subsets:
  {2}  {2}    {2}    {2}
       {3}    {3}    {3}
       {1,3}  {4}    {4}
              {1,3}  {5}
              {1,4}  {1,3}
              {3,4}  {1,4}
                     {1,5}
                     {3,4}
                     {3,5}
                     {4,5}
                     {1,4,5}
                     {2,4,5}

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
A000009 counts subsets summing to n.
A000124 counts distinct possible sums of subsets of {1..n}.
A229816 counts partitions whose length is not a part, complement A002865.
A007865/A085489/A151897 count certain types of sum-free subsets.
A088809/A093971/A364534 count certain types of sum-full subsets.
A124506 appears to count combination-free subsets, differences of A326083.
A237667 counts sum-free partitions, ranks A364531.
Triangles:
A046663 counts partitions of n without a subset-sum k, strict A365663.
A365381 counts sets with a subset summing to k, without A366320.
A365541 counts sets containing two distinct elements summing to k.
Cf. A068911, A103580, A240861, A288728, A326080, A326083, A364346, A364349, A365046, A365376, A365377.

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

a(n) = 2^n - A367216(n). - Chai Wah Wu, Nov 14 2023

a(16)-a(28) from Chai Wah Wu, Nov 14 2023

a(29)-a(35) from Max Alekseyev, Feb 25 2025

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}]

cp's OEIS Frontend

A365663 Triangle read by rows where T(n,k) is the number of strict integer partitions of n without a subset summing to k.

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A124506 Number of numerical semigroups with Frobenius number n; that is, numerical semigroups for which the largest integer not belonging to them is n.

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A364346 Number of strict integer partitions of n such that there is no ordered triple of parts (a,b,c) (repeats allowed) satisfying a + b = c. A variation of sum-free strict partitions.

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A367213 Number of integer partitions of n whose length (number of parts) is not equal to the sum of any submultiset.

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A367214 Number of strict integer partitions of n whose length (number of parts) is equal to the sum of some submultiset.

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A367215 Number of strict integer partitions of n whose length (number of parts) is not equal to the sum of any subset.

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A367216 Number of subsets of {1..n} whose cardinality is equal to the sum of some subset.

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A367217 Number of subsets of {1..n} whose cardinality is not equal to the sum of any subset.

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