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.

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A365311 Number of strict integer partitions with sum <= n that can be linearly combined using nonnegative coefficients to obtain n.

Original entry on oeis.org

0, 1, 2, 3, 5, 6, 11, 12, 20, 24, 35, 38, 63, 63, 92, 112, 148, 160, 230, 244, 339, 383, 478, 533, 726, 781, 978, 1123, 1394, 1526, 1960, 2112, 2630, 2945, 3518, 3964, 4856, 5261, 6307, 7099, 8464, 9258, 11140, 12155, 14419, 16093, 18589, 20565, 24342, 26597, 30948
Offset: 0

Views

Author

Gus Wiseman, Sep 04 2023

Keywords

Examples

			The strict partition (6,3) cannot be linearly combined to obtain 10, so is not counted under a(10).
The strict partition (4,2) has 6 = 1*4 + 1*2 so is counted under a(6), but (4,2) cannot be linearly combined to obtain 7 so is not counted under a(7).
The a(1) = 1 through a(7) = 12 strict partitions:
  (1)  (1)  (1)    (1)    (1)    (1)      (1)
       (2)  (3)    (2)    (5)    (2)      (7)
            (2,1)  (4)    (2,1)  (3)      (2,1)
                   (2,1)  (3,1)  (6)      (3,1)
                   (3,1)  (3,2)  (2,1)    (3,2)
                          (4,1)  (3,1)    (4,1)
                                 (3,2)    (4,3)
                                 (4,1)    (5,1)
                                 (4,2)    (5,2)
                                 (5,1)    (6,1)
                                 (3,2,1)  (3,2,1)
                                          (4,2,1)

Links

Crossrefs

For positive coefficients we have A088314.
The positive complement is counted by A088528.
The version for subsets is A365073.
The complement is counted by A365312.
For non-strict partitions we have 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. A093971, A237113, A237668, A326080, A363225, A364272, A364534, A364914, A365043, A365314, 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[Select[Join@@Array[IntegerPartitions,n],UnsameQ@@#&],combs[n,#]!={}&]],{n,10}]
  • Python
    from math import isqrt
from sympy.utilities.iterables import partitions
def A365311(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,m=isqrt(1+(n<<3))>>1) if max(b.values()) == 1 and any(set(d).issubset(set(b)) for d in a)) # Chai Wah Wu, Sep 13 2023

Extensions

a(26)-a(50) from Chai Wah Wu, Sep 13 2023

A365377 Number of subsets of {1..n} without a subset summing to n.

Original entry on oeis.org

0, 1, 2, 3, 6, 9, 17, 26, 49, 72, 134, 201, 366, 544, 984, 1436, 2614, 3838, 6770, 10019, 17767, 25808, 45597, 66671, 116461, 169747, 295922, 428090, 750343, 1086245, 1863608, 2721509, 4705456, 6759500, 11660244, 16877655, 28879255, 41778027, 71384579, 102527811, 176151979
Offset: 0

Views

Author

Gus Wiseman, Sep 08 2023

Keywords

Examples

			The a(1) = 1 through a(6) = 17 subsets:
  {}  {}   {}   {}     {}     {}
      {1}  {1}  {1}    {1}    {1}
           {2}  {2}    {2}    {2}
                {3}    {3}    {3}
                {1,2}  {4}    {4}
                {2,3}  {1,2}  {5}
                       {1,3}  {1,2}
                       {2,4}  {1,3}
                       {3,4}  {1,4}
                              {2,3}
                              {2,5}
                              {3,4}
                              {3,5}
                              {4,5}
                              {1,3,4}
                              {2,3,5}
                              {3,4,5}

Links

Crossrefs

The complement w/ re-usable parts is A365073.
The complement is counted by A365376.
The version with re-usable parts is A365380.
A000009 counts sets summing to n, multisets A000041.
A000124 counts distinct possible sums of subsets of {1..n}.
A124506 appears to count combination-free subsets, differences of A326083.
A364350 counts combination-free strict partitions, complement A364839.
A365046 counts combination-full subsets, differences of A364914.
A365381 counts subsets of {1..n} with a subset summing to k.
Cf. A007865, A085489, A088809, A093971, A103580, A151897, A236912, A237113, A237668, A326080, A364349, A364534.

Programs

  • Mathematica
    Table[Length[Select[Subsets[Range[n]], FreeQ[Total/@Subsets[#],n]&]],{n,0,10}]
  • PARI
    isok(s, n) = forsubset(#s, ss, if (vecsum(vector(#ss, k, s[ss[k]])) == n, return(0))); return(1);
a(n) = my(nb=0); forsubset(n, s, if (isok(s, n), nb++)); nb; \\ Michel Marcus, Sep 09 2023
  • Python
    from itertools import combinations, chain
from sympy.utilities.iterables import partitions
def A365377(n):
    if n == 0: return 0
    nset = set(range(1,n+1))
    s, c = [set(p) for p in partitions(n,m=n,k=n) if max(p.values(),default=1) == 1], 1
    for a in chain.from_iterable(combinations(nset,m) for m in range(2,n+1)):
        if sum(a) >= n:
            aset = set(a)
            for p in s:
                if p.issubset(aset):
                    c += 1
                    break
    return (1<Chai Wah Wu, Sep 09 2023

Formula

a(n) = 2^n-A365376(n). - Chai Wah Wu, Sep 09 2023

Extensions

a(16)-a(27) from Michel Marcus, Sep 09 2023
a(28)-a(32) from Chai Wah Wu, Sep 09 2023
a(33)-a(35) from Chai Wah Wu, Sep 10 2023
More terms from David A. Corneth, Sep 10 2023

A364532 Positive integers with a prime index equal to the sum of prime indices of some nonprime divisor. Heinz numbers of a variation of sum-full partitions.

Original entry on oeis.org

12, 24, 30, 36, 40, 48, 60, 63, 70, 72, 80, 84, 90, 96, 108, 112, 120, 126, 132, 140, 144, 150, 154, 156, 160, 165, 168, 180, 189, 192, 198, 200, 204, 210, 216, 220, 224, 228, 240, 252, 264, 270, 273, 276, 280, 286, 288, 300, 308, 312, 315, 320, 324, 325, 330
Offset: 1

Views

Author

Gus Wiseman, Aug 01 2023

Keywords

Comments

First differs from A299729 (non-knapsack) in lacking 525: {2,3,3,4}.
First differs from A325777 in having 462: {1,2,4,5} and lacking 675:{2,2,2,3,3}.
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
These are the Heinz numbers of partitions containing the sum of some non-singleton submultiset.

Examples

			The terms together with their prime indices begin:
  12: {1,1,2}
  24: {1,1,1,2}
  30: {1,2,3}
  36: {1,1,2,2}
  40: {1,1,1,3}
  48: {1,1,1,1,2}
  60: {1,1,2,3}
  63: {2,2,4}
  70: {1,3,4}
  72: {1,1,1,2,2}
  80: {1,1,1,1,3}
  84: {1,1,2,4}
  90: {1,2,2,3}
  96: {1,1,1,1,1,2}

Crossrefs

Partitions not of this type are counted by A237667, strict A364349.
Partitions of this type are counted by A237668, strict A364272.
The binary complement is A364461, re-usable A364347 (counted by A364345).
The binary version is A364462, re-usable A364348 (counted by A363225).
The complement is A364531.
Subsets of this type are counted by A364534, complement A151897.
A000005 counts divisors, nonprime A033273, composite A055212.
A001222 counts prime indices.
A108917 counts knapsack partitions, strict A275972, for subsets A325864.
A112798 lists prime indices, sum A056239.
A299701 counts distinct subset-sums of prime indices.
A299702 ranks knapsack partitions, complement A299729.
Cf. A085489, A088809, A093971, A236912, A237113, A301900, A326083, A364350.

Programs

  • Mathematica
    Select[Range[100],Intersection[prix[#],Total/@Subsets[prix[#],{2,Length[prix[#]]}]]!={}&]

A364755 Number of subsets of {1..n} containing n but not containing the sum of any two distinct elements.

Original entry on oeis.org

0, 1, 2, 3, 6, 9, 15, 24, 41, 60, 99, 149, 236, 355, 552, 817, 1275, 1870, 2788, 4167, 6243, 9098, 13433, 19718, 28771, 42137, 60652, 88603, 127555, 185200, 261781, 382931, 541022, 783862, 1096608, 1595829, 2217467, 3223064, 4441073, 6465800, 8893694
Offset: 0

Views

Author

Gus Wiseman, Aug 11 2023

Keywords

Examples

			The subset S = {1,3,6,8} has pair-sums {4,7,9,11,14}, which are disjoint from S, so it is counted under a(8).
The a(1) = 1 through a(6) = 15 subsets:
  {1}  {2}    {3}    {4}      {5}      {6}
       {1,2}  {1,3}  {1,4}    {1,5}    {1,6}
              {2,3}  {2,4}    {2,5}    {2,6}
                     {3,4}    {3,5}    {3,6}
                     {1,2,4}  {4,5}    {4,6}
                     {2,3,4}  {1,2,5}  {5,6}
                              {1,3,5}  {1,2,6}
                              {2,4,5}  {1,3,6}
                              {3,4,5}  {1,4,6}
                                       {2,3,6}
                                       {2,5,6}
                                       {3,4,6}
                                       {3,5,6}
                                       {4,5,6}
                                       {3,4,5,6}

Links

Crossrefs

Partial sums are A085489(n) - 1, complement counted by A364534.
With re-usable parts we have A288728.
The complement with n is counted by A364756, first differences of A088809.
Cf. A007865, A050291, A054519, A093971, A151897, A236912, A326020, A326080, A326083, A364272, A364349, A364533.

Programs

  • Mathematica
    Table[Length[Select[Subsets[Range[n]],MemberQ[#,n]&&Intersection[#,Total/@Subsets[#,{2}]]=={}&]],{n,0,10}]

Formula

First differences of A085489.

Extensions

a(21) onwards added (using A085489) by Andrew Howroyd, Jan 13 2024

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

Views

Author

Gus Wiseman, Aug 03 2023

Keywords

Examples

			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

Crossrefs

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.

Programs

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

A364756 Number of subsets of {1..n} containing n and some element equal to the sum of two distinct others.

Original entry on oeis.org

0, 0, 0, 1, 2, 7, 17, 40, 87, 196, 413, 875, 1812, 3741, 7640, 15567, 31493, 63666, 128284, 257977, 518045, 1039478, 2083719, 4174586, 8359837, 16735079, 33493780, 67020261, 134090173, 268250256, 536609131, 1073358893, 2146942626, 4294183434, 8588837984, 17178273355
Offset: 0

Views

Author

Gus Wiseman, Aug 11 2023

Keywords

Examples

			The subset S = {1,3,6,8} has pair-sums {4,7,9,11,14}, which are disjoint from S, so it is not counted under a(8).
The subset {2,3,4,6} has pair-sum 2 + 4 = 6, so is counted under a(6).
The a(0) = 0 through a(6) = 17 subsets:
  .  .  .  {1,2,3}  {1,3,4}    {1,4,5}      {1,5,6}
                    {1,2,3,4}  {2,3,5}      {2,4,6}
                               {1,2,3,5}    {1,2,3,6}
                               {1,2,4,5}    {1,2,4,6}
                               {1,3,4,5}    {1,2,5,6}
                               {2,3,4,5}    {1,3,4,6}
                               {1,2,3,4,5}  {1,3,5,6}
                                            {1,4,5,6}
                                            {2,3,4,6}
                                            {2,3,5,6}
                                            {2,4,5,6}
                                            {1,2,3,4,6}
                                            {1,2,3,5,6}
                                            {1,2,4,5,6}
                                            {1,3,4,5,6}
                                            {2,3,4,5,6}
                                            {1,2,3,4,5,6}

Links

Crossrefs

Partial sums are A088809, non-binary A364534.
With re-usable parts we have differences of A093971, complement A288728.
The complement with n is counted by A364755, partial sums A085489(n) - 1.
Cf. A000079, A007865, A050291, A051026, A103580, A151897, A236912, A326080, A326083, A364272.

Programs

  • Mathematica
    Table[Length[Select[Subsets[Range[n]],MemberQ[#,n]&&Intersection[#,Total/@Subsets[#,{2}]]!={}&]],{n,0,10}]

Formula

First differences of A088809.

Extensions

a(16) onwards added (using A088809) by Andrew Howroyd, Jan 13 2024

A365320 Number of pairs of distinct positive integers <= n that cannot be linearly combined with nonnegative coefficients to obtain n.

Original entry on oeis.org

0, 0, 0, 0, 0, 2, 1, 7, 5, 12, 12, 27, 14, 42, 36, 47, 47, 83, 58, 109, 80, 116, 126, 172, 111, 195, 192, 219, 202, 294, 210, 342, 286, 354, 369, 409, 324, 509, 480, 523, 452, 640, 507, 711, 622, 675, 747, 865, 654, 916, 842, 964, 922, 1124, 940, 1147, 1029
Offset: 0

Views

Author

Gus Wiseman, Sep 06 2023

Keywords

Comments

Are there only two cases of nonzero adjacent equal parts, at positions n = 9, 15?

Examples

			The pair p = (3,6) cannot be linearly combined to obtain 8 or 10, so p is counted under a(8) and a(10), but we have 9 = 1*3 + 1*6 or 9 = 3*3 + 0*6, so p not counted under a(9).
The a(5) = 2 through a(10) = 12 pairs:
  (2,4)  (4,5)  (2,4)  (3,6)  (2,4)  (3,6)
  (3,4)         (2,6)  (3,7)  (2,6)  (3,8)
                (3,5)  (5,6)  (2,8)  (3,9)
                (3,6)  (5,7)  (4,6)  (4,7)
                (4,5)  (6,7)  (4,7)  (4,8)
                (4,6)         (4,8)  (4,9)
                (5,6)         (5,6)  (6,7)
                              (5,7)  (6,8)
                              (5,8)  (6,9)
                              (6,7)  (7,8)
                              (6,8)  (7,9)
                              (7,8)  (8,9)

Crossrefs

The unrestricted version is A000217, ranks A001358.
For strict partitions we have A365312, complement A365311.
The (binary) complement is A365314, positive A365315.
The case of positive coefficients is A365321, for all subsets A365322.
For partitions we have A365378, complement A365379.
For all subsets instead of just pairs we have A365380, complement A365073.
A004526 counts partitions of length 2, shift right for strict.
A007865 counts sum-free subsets, complement A093971.
A179822 and A326080 count sum-closed subsets.
A326083 and A124506 appear to count combination-free subsets.
A364350 counts combination-free strict partitions.
A364914 and A365046 count combination-full subsets.
Cf. A070880, A088314, A088809, A151897, A326020, A364839.

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[Subsets[Range[n],{2}],combs[n,#]=={}&]],{n,0,30}]
  • Python
    from itertools import count
from sympy import divisors
def A365320(n):
    a = set()
    for i in range(1,n+1):
        if not n%i:
            a.update(tuple(sorted((i,j))) for j in range(1,n+1) if j!=i)
        else:
            for j in count(0,i):
                if j > n:
                    break
                k = n-j
                for d in divisors(k):
                    if d>=i:
                        break
                    a.add((d,i))
    return (n*(n-1)>>1)-len(a) # Chai Wah Wu, Sep 13 2023

A364466 Number of subsets of {1..n} where some element is a difference of two consecutive elements.

Original entry on oeis.org

0, 0, 1, 2, 6, 14, 34, 74, 164, 345, 734, 1523, 3161, 6488, 13302, 27104, 55150, 111823, 226443, 457586, 923721, 1862183, 3751130, 7549354, 15184291, 30521675, 61322711, 123151315, 247230601, 496158486, 995447739, 1996668494, 4004044396, 8027966324, 16092990132, 32255168125
Offset: 0

Views

Author

Gus Wiseman, Jul 31 2023

Keywords

Comments

In other words, the elements are not disjoint from their own first differences.

Examples

			The a(0) = 0 through a(5) = 14 subsets:
  .  .  {1,2}  {1,2}    {1,2}      {1,2}
               {1,2,3}  {2,4}      {2,4}
                        {1,2,3}    {1,2,3}
                        {1,2,4}    {1,2,4}
                        {1,3,4}    {1,2,5}
                        {1,2,3,4}  {1,3,4}
                                   {1,4,5}
                                   {2,3,5}
                                   {2,4,5}
                                   {1,2,3,4}
                                   {1,2,3,5}
                                   {1,2,4,5}
                                   {1,3,4,5}
                                   {1,2,3,4,5}

Crossrefs

For differences of all pairs we have A093971, complement A196723.
For partitions we have A363260, complement A364467.
The complement is counted by A364463.
For subset-sums instead of differences we have A364534, complement A325864.
For strict partitions we have A364536, complement A364464.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A050291 counts double-free subsets, complement A088808.
A108917 counts knapsack partitions, strict A275972.
A325325 counts partitions with all distinct differences, strict A320347.
Cf. A011782, A212986, A237113, A325877, A325878, A326083, A363225.

Programs

  • Mathematica
    Table[Length[Select[Subsets[Range[n]],Intersection[#,Differences[#]]!={}&]],{n,0,10}]
  • Python
    from itertools import combinations
def A364466(n): return sum(1 for l in range(n+1) for c in combinations(range(1,n+1),l) if not set(c).isdisjoint({c[i+1]-c[i] for i in range(l-1)})) # Chai Wah Wu, Sep 26 2023

Formula

a(n) = 2^n - A364463(n). - Chai Wah Wu, Sep 26 2023

Extensions

a(21)-a(32) from Chai Wah Wu, Sep 26 2023
a(33)-a(35) from Chai Wah Wu, Sep 27 2023

A365544 Number of subsets of {1..n} containing two distinct elements summing to n.

Original entry on oeis.org

0, 0, 0, 2, 4, 14, 28, 74, 148, 350, 700, 1562, 3124, 6734, 13468, 28394, 56788, 117950, 235900, 484922, 969844, 1979054, 3958108, 8034314, 16068628, 32491550, 64983100, 131029082, 262058164, 527304974, 1054609948, 2118785834, 4237571668, 8503841150, 17007682300
Offset: 0

Views

Author

Gus Wiseman, Sep 20 2023

Keywords

Examples

			The a(1) = 0 through a(5) = 14 subsets:
  .  .  {1,2}    {1,3}      {1,4}
        {1,2,3}  {1,2,3}    {2,3}
                 {1,3,4}    {1,2,3}
                 {1,2,3,4}  {1,2,4}
                            {1,3,4}
                            {1,4,5}
                            {2,3,4}
                            {2,3,5}
                            {1,2,3,4}
                            {1,2,3,5}
                            {1,2,4,5}
                            {1,3,4,5}
                            {2,3,4,5}
                            {1,2,3,4,5}

Links

Crossrefs

For strict partitions we have A140106 shifted left.
The version for partitions is A004526.
The complement is counted by A068911.
For all subsets of elements we have A365376.
Main diagonal k = n of A365541.
A000009 counts subsets summing to n.
A007865/A085489/A151897 count certain types of sum-free subsets.
A093971/A088809/A364534 count certain types of sum-full subsets.
A365381 counts subsets with a subset summing to k.
Cf. A008967, A095944, A167762, A238628, A288728, A326083, A364272, A365377.

Programs

  • Mathematica
    Table[Length[Select[Subsets[Range[n]],MemberQ[Total/@Subsets[#,{2}],n]&]],{n,0,10}]
  • Python
    def A365544(n): return (1<>1)<<1 if n&1 else 3**(n-1>>1)<<2) if n else 0 # Chai Wah Wu, Aug 30 2024

Formula

a(n) = 2^n - A068911(n).
From Alois P. Heinz, Aug 30 2024: (Start)
G.f.: 2*x^3/((2*x-1)*(3*x^2-1)).
a(n) = 2 * A167762(n-1) for n>=1. (End)

A364467 Number of integer partitions of n where some part is the difference of two consecutive parts.

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 4, 5, 9, 13, 21, 28, 42, 55, 78, 106, 144, 187, 255, 325, 429, 554, 717, 906, 1165, 1460, 1853, 2308, 2899, 3582, 4468, 5489, 6779, 8291, 10173, 12363, 15079, 18247, 22124, 26645, 32147, 38555, 46285, 55310, 66093, 78684, 93674, 111104
Offset: 0

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Author

Gus Wiseman, Jul 31 2023

Keywords

Comments

In other words, the parts are not disjoint from their own first differences.

Examples

			The a(3) = 1 through a(9) = 13 partitions:
  (21)  (211)  (221)   (42)     (421)     (422)      (63)
               (2111)  (321)    (2221)    (431)      (621)
                       (2211)   (3211)    (521)      (3321)
                       (21111)  (22111)   (3221)     (4221)
                                (211111)  (4211)     (4311)
                                          (22211)    (5211)
                                          (32111)    (22221)
                                          (221111)   (32211)
                                          (2111111)  (42111)
                                                     (222111)
                                                     (321111)
                                                     (2211111)
                                                     (21111111)

Crossrefs

For all differences of pairs parts we have A363225, complement A364345.
The complement is counted by A363260.
For subsets of {1..n} we have A364466, complement A364463.
The strict case is A364536, complement A364464.
These partitions have ranks A364537.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A050291 counts double-free subsets, complement A088808.
A323092 counts double-free partitions, ranks A320340.
A325325 counts partitions with distinct first differences.
Cf. A002865, A025065, A093971, A108917, A196723, A229816, A236912, A237113, A237667, A320347, A326083.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],Intersection[#,-Differences[#]]!={}&]],{n,0,30}]
  • Python
    from collections import Counter
from sympy.utilities.iterables import partitions
def A364467(n): return sum(1 for s,p in map(lambda x: (x[0],tuple(sorted(Counter(x[1]).elements()))), partitions(n,size=True)) if not set(p).isdisjoint({p[i+1]-p[i] for i in range(s-1)})) # Chai Wah Wu, Sep 26 2023
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