A317142 -id:A317142 - cp's OEIS frontend

A384390 Heinz numbers of integer partitions with a unique proper way to choose disjoint strict partitions of each part.

5, 7, 21, 22, 26, 33, 35, 39, 102, 114, 130, 154, 165, 170, 190, 195, 231, 238, 255, 285

Offset: 1

Gus Wiseman, Jun 02 2025

nonn

By "proper" we exclude the case of all singletons, which is disjoint in the strict case.

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

			The strict partition (7,2,1) with Heinz number 102 can only be properly refined as ((4,3),(2),(1)), so 102 is in the sequence. The other refinement ((7),(2),(1)) is not proper.
The terms together with their prime indices begin:
    5: {3}
    7: {4}
   21: {2,4}
   22: {1,5}
   26: {1,6}
   33: {2,5}
   35: {3,4}
   39: {2,6}
  102: {1,2,7}
  114: {1,2,8}
  130: {1,3,6}
  154: {1,4,5}
  165: {2,3,5}
  170: {1,3,7}
  190: {1,3,8}
  195: {2,3,6}
  231: {2,4,5}
  238: {1,4,7}
  255: {2,3,7}
  285: {2,3,8}

The non-proper version is A383707, counted by A179009.
Partitions of this type are counted by A384319, non-strict A384323 (ranks A384347).
This is the unique case of A384321, counted by A384317.
This is the case of a unique proper choice in A384322.
The complement is A384349 \/ A384393.
These are positions of 1 in A384389.
A048767 is the Look-and-Say transform, fixed points A048768, counted by A217605.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
A239455 counts Look-and-Say or section-sum partitions, ranks A351294 or A381432.
A351293 counts non-Look-and-Say or non-section-sum partitions, ranks A351295 or A381433.
A357982 counts strict partitions of each prime index, non-strict A299200.
Cf. A382912, counted by A383710, odd case A383711.
Cf. A382913, counted by A383708, odd case A383533.
Cf. A098859, A122111, A130091, A279375, A279790, A317142, A351201, A381454, A384005, A384320.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
pofprop[y_]:=Select[DeleteCases[Join@@@Tuples[IntegerPartitions/@y],y],UnsameQ@@#&];
Select[Range[100],Length[pofprop[prix[#]]]==1&]

A382079 Number of integer partitions of n that can be partitioned into a set of sets in exactly one way.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 3, 4, 6, 5, 10, 9, 13, 14, 21, 20, 32, 31, 42, 47, 63, 62, 90, 94, 117, 138, 170, 186, 235, 260, 315, 363, 429, 493, 588, 674, 795, 901, 1060, 1209, 1431, 1608, 1896, 2152, 2515, 2854, 3310, 3734, 4368, 4905, 5686

Offset: 0

Gus Wiseman, Mar 20 2025

			The unique multiset partition for (3222111) is {{1},{2},{1,2},{1,2,3}}.
The a(1) = 1 through a(12) = 13 partitions:
  1  2  3  4    5    6     7    8      9      A      B      C
           211  221  411   322  332    441    433    443    552
                311  2211  331  422    522    442    533    633
                           511  611    711    622    551    822
                                3311   42111  811    722    A11
                                32111         3322   911    4422
                                              4411   42221  5511
                                              32221  53111  33321
                                              43111  62111  52221
                                              52111         54111
                                                            63111
                                                            72111
                                                            3222111

Normal multiset partitions of this type are counted by A116539, see A381718.
These partitions are ranked by A293511.
MM-numbers of these multiset partitions (sets of sets) are A302494, see A302478, A382201.
Twice-partitions of this type (sets of sets) are counted by A358914, see A279785.
For at least one choice we have A382077 (ranks A382200), see A381992 (ranks A382075).
For no choices we have A382078 (ranks A293243), see A381990 (ranks A381806).
For distinct block-sums instead of blocks we have A382460, ranked by A381870.
Set multipartitions: A089259, A116540, A270995, A296119, A318360.
A000041 counts integer partitions, strict A000009.
A050320 counts multiset partitions of prime indices into sets.
A050326 counts multiset partitions of prime indices into distinct sets, see A381633.
A265947 counts refinement-ordered pairs of integer partitions.
Cf. A002846, A213427, A299202, A317142, A381078, A381441, A381454, A381636.

ssfacs[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#,d]&)/@Select[ssfacs[n/d],Min@@#>d&],{d,Select[Rest[Divisors[n]],SquareFreeQ]}]];
Table[Length[Select[IntegerPartitions[n],Length[ssfacs[Times@@Prime/@#]]==1&]],{n,0,15}]

a(21)-a(50) from Bert Dobbelaere, Mar 29 2025

A384319 Number of strict integer partitions of n with exactly two possible ways to choose disjoint strict partitions of each part.

Original entry on oeis.org

0, 0, 0, 1, 1, 0, 2, 3, 1, 0, 4, 4, 4, 2, 0, 6, 7, 8, 8, 3, 2, 9, 9, 14, 13, 6, 7, 3, 15, 13, 20

Offset: 0

Gus Wiseman, May 28 2025

			For y = (5,4,2) we have choices ((5),(4),(2)) and ((5),(3,1),(2)), so y is counted under a(11).
The a(3) = 1 through a(11) = 4 partitions:
  (3)  (4)  .  (4,2)  (4,3)  (6,2)  .  (5,3,2)  (5,4,2)
               (5,1)  (5,2)            (5,4,1)  (6,3,2)
                      (6,1)            (6,3,1)  (7,3,1)
                                       (7,2,1)  (8,2,1)

The case of a unique choice is A179009, ranks A383707.
Choices of this type for each prime index are counted by A383706.
The non-strict version for at least one choice is A383708, ranks A382913.
The non-strict version for no choices is A383710, ranks A382912.
The non-strict version for more than one choice is A384317, ranks A384321.
The version for at least one choice is A384322, counted by A384318.
The non-strict version is A384323, ranks A384347.
These partitions are ranked by A384390.
A239455 counts Look-and-Say or section-sum partitions, ranks A351294 or A381432.
A351293 counts non Look-and-Say or non section-sum partitions, ranks A351295 or A381433.
Cf. A098859, A279375, A299200, A317142, A357982, A381454, A383533, A383711, A384320.

Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Length[pof[#]]==2&]],{n,0,30}]

A384323 Number of integer partitions of n with exactly two possible ways to choose disjoint strict partitions of each part.

Original entry on oeis.org

0, 0, 0, 1, 1, 0, 3, 3, 2, 0, 6, 6, 6, 6, 4, 10, 10, 14, 16, 15, 16, 17, 20, 25, 27, 28, 37, 43, 31, 42, 44

Offset: 0

Gus Wiseman, May 30 2025

			For y = (4,3,3) we have two ways: ((4),(3),(2,1)) and ((4),(2,1),(3)), so y is counted under a(10).
The a(0) = 0 through a(15) = 10 partitions:
  .  .  .  3  4  .  33  43  44  .  433  533  543  544  554  5433
                    42  52  62     442  542  552  553  644  5442
                    51  61         532  551  633  652  662  5532
                                   541  632  732  661  833  5541
                                   631  731  741  733       6432
                                   721  821  831  832       6531
                                                            7431
                                                            7521
                                                            8421
                                                            9321

For just one choice we have A179009, ranked by A383707.
Twice-partitions of this type are counted by A279790.
For at least one choice we have A383708, odd case A383533.
For no choices we have A383710, odd case A383711.
For more than one choice we have A384317, ranked by A384321.
The strict version for at least one choice is A384318, ranked by A384322.
The strict version is A384319, ranked by A384390.
These partitions are ranked by A384347 = positions of 2 in A383706.
A000041 counts integer partitions, strict A000009.
A048767 is the Look-and-Say transform, fixed points A048768, counted by A217605.
A239455 counts Look-and-Say partitions, ranks A351294 or A381432.
A351293 counts non-Look-and-Say partitions, ranks A351295 or A381433.
A357982 counts choices of strict partitions of each prime index.
Cf. A098859, A299200, A317142, A381454, A382525, A382912, A382913, A384005.

pof[y_]:=Select[Join@@@Tuples[IntegerPartitions/@y],UnsameQ@@#&];
Table[Length[Select[IntegerPartitions[n],Length[pof[#]]==2&]],{n,0,15}]

A384349 Heinz numbers of integer partitions with no proper way to choose disjoint strict partitions of each part.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 24, 27, 28, 30, 32, 36, 40, 42, 44, 45, 48, 50, 52, 54, 56, 60, 63, 64, 66, 68, 70, 72, 75, 76, 78, 80, 81, 84, 88, 90, 92, 96, 98, 99, 100, 104, 105, 108, 110, 112, 116, 117, 120, 124, 125, 126, 128, 132, 135

Offset: 1

Gus Wiseman, Jun 03 2025

nonn

By "proper" we exclude the case of all singletons, which is disjoint when n is squarefree.

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

			The prime indices of 102 are {1,2,7}, which has proper disjoint choice ((1),(2),(3,4)), so 102 is not in the sequence.
The terms together with their prime indices begin:
     1: {}           27: {2,2,2}        63: {2,2,4}
     2: {1}          28: {1,1,4}        64: {1,1,1,1,1,1}
     3: {2}          30: {1,2,3}        66: {1,2,5}
     4: {1,1}        32: {1,1,1,1,1}    68: {1,1,7}
     6: {1,2}        36: {1,1,2,2}      70: {1,3,4}
     8: {1,1,1}      40: {1,1,1,3}      72: {1,1,1,2,2}
     9: {2,2}        42: {1,2,4}        75: {2,3,3}
    10: {1,3}        44: {1,1,5}        76: {1,1,8}
    12: {1,1,2}      45: {2,2,3}        78: {1,2,6}
    14: {1,4}        48: {1,1,1,1,2}    80: {1,1,1,1,3}
    15: {2,3}        50: {1,3,3}        81: {2,2,2,2}
    16: {1,1,1,1}    52: {1,1,6}        84: {1,1,2,4}
    18: {1,2,2}      54: {1,2,2,2}      88: {1,1,1,5}
    20: {1,1,3}      56: {1,1,1,4}      90: {1,2,2,3}
    24: {1,1,1,2}    60: {1,1,2,3}      92: {1,1,9}

The non-proper version appears to be A382912, counted by A383710.
The non-proper complement appears to be A382913, counted by A383708.
The complement is A384321, counted by A384317.
These partitions are counted by A384348.
These are the positions of 0 in A384389.
The case of a unique proper choice is A384390, counted by A384319.
A048767 is the Look-and-Say transform, fixed points A048768.
A056239 adds up prime indices, row sums of A112798.
A179009 counts maximally refined strict partitions, ranks A383707.
A279790 and A279375 count ways to choose disjoint strict partitions of prime indices.
Cf. A122111, A130091, A317142, A326080, A351294, A357982, A381454, A382525, A383706, A384320, A384322.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
pofprop[y_]:=Select[DeleteCases[Join@@@Tuples[IntegerPartitions/@y],y],UnsameQ@@#&];
Select[Range[100],Length[pofprop[prix[#]]]==0&]

A381637 Number of multisets that can be obtained by taking the sum of each block of a multiset partition of the prime indices of n into blocks with distinct sums.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 2, 2, 1, 3, 1, 3, 2, 2, 1, 3, 1, 2, 2, 3, 1, 4, 1, 3, 2, 2, 2, 4, 1, 2, 2, 4, 1, 5, 1, 3, 3, 2, 1, 4, 1, 3, 2, 3, 1, 5, 2, 5, 2, 2, 1, 5, 1, 2, 2, 4, 2, 5, 1, 3, 2, 4, 1, 5, 1, 2, 3, 3, 2, 5, 1, 5, 2, 2, 1, 6, 2, 2, 2

Offset: 1

Gus Wiseman, Mar 10 2025

nonn

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.

			The prime indices of 84 are {1,1,2,4}, with 7 multiset partitions into blocks with distinct sums:
  {{1,1,2,4}}
  {{1},{1,2,4}}
  {{2},{1,1,4}}
  {{1,1},{2,4}}
  {{1,2},{1,4}}
  {{1},{2},{1,4}}
  {{1},{4},{1,2}}
with block-sums: {8}, {1,7}, {2,6}, {2,6}, {3,5}, {1,2,5}, {1,3,4}, of which 6 are distinct, so a(84) = 6.

Allowing any block-sums gives A317141  (lower A300383), before sums A001055.
Before taking sums we had A321469.
For distinct blocks instead of distinct block-sums we have A381452.
If each block is a set we have A381634 (zeros A381806), before sums A381633.
For equal instead of distinct block-sums we have A381872, before sums A321455.
Other multiset partitions of prime indices:
- For multisets of constant multisets (A000688) see A381455 (upper), A381453 (lower).
- For set multipartitions (A050320) see A381078 (upper), A381454 (lower).
- For sets of constant multisets (A050361) see A381715.
- For sets of constant multisets with distinct sums (A381635) see A381716, A381636.
A003963 gives product of prime indices.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
A265947 counts refinement-ordered pairs of integer partitions.
Cf. A000720, A001222, A001970, A002846, A045778, A066328, A213385, A299200, A299201, A299202, A300385, A317142.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[mset_]:=Union[Sort[Sort/@(#/.x_Integer:>mset[[x]])]&/@sps[Range[Length[mset]]]];
Table[Length[Union[Sort[Total/@#]&/@Select[mps[prix[n]],UnsameQ@@Total/@#&]]],{n,100}]

A382460 Number of integer partitions of n that can be partitioned into sets with distinct sums in exactly one way.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 3, 4, 6, 5, 10, 10, 13, 15, 22, 20, 32, 32, 43, 49, 65, 64, 92, 96, 121, 140, 173, 192

Offset: 0

Gus Wiseman, Mar 29 2025

			The partition y = (3,3,2,1,1,1) has 2 partitions into sets: {{1},{3},{1,2},{1,3}} and {{1},{1,3},{1,2,3}}, but only the latter has distinct sums, so y is counted under a(11)
The a(1) = 1 through a(10) = 10 partitions (A=10):
  1  2  3  4    5    6     7    8      9      A
           211  221  411   322  332    441    433
                311  2211  331  422    522    442
                           511  611    711    622
                                3311   42111  811
                                32111         3322
                                              4411
                                              32221
                                              43111
                                              52111

Twice-partitions of this type are counted by A279785.
Multiset partitions of this type are counted by A381633.
Normal multiset partitions of this type are counted by A381718.
These partitions are ranked by A381870.
For no choices we have A381990, ranks A381806, see A382078, ranks A293243.
For at least one choice we have A381992, ranks A382075, see A382077, ranks A382200.
For distinct blocks instead of block-sums we have A382079, ranks A293511.
MM-numbers of these multiset partitions are A382201, see A302478.
For constant instead of strict blocks we have A382301, ranks A381991.
Set systems: A050320, A050326, A050342, A116539, A296120, A318361.
Set multipartitions: A089259, A116540, A270995, A296119, A318360.
A000041 counts integer partitions, strict A000009.
A265947 counts refinement-ordered pairs of integer partitions.
Cf. A002846, A047966, A213427, A299202, A317142, A358914, A381441, A381454, A381636.

hwt[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]*k]];
ssfacs[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#,d]&) /@ Select[ssfacs[n/d],Min@@#>d&],{d,Select[Rest[Divisors[n]],SquareFreeQ]}]];
Table[Length[Select[IntegerPartitions[n], Length[Select[ssfacs[Times@@Prime/@#],UnsameQ@@hwt/@#&]]==1&]],{n,0,15}]

A381994 Number of integer partitions of n that cannot be partitioned into sets with equal sums.

Original entry on oeis.org

0, 0, 0, 0, 1, 3, 3, 9, 12, 17, 27, 43, 46, 82, 103, 133, 181, 258, 295

Offset: 0

Gus Wiseman, Mar 17 2025

nonn

			For y = (3,3,1,1) we have {{1,3},{1,3}}, so y is not counted under a(8).
For y = (3,2,2,1), although we have {{1,3},{2,2}}, the block {2,2} is not a set, so y is counted under a(8).
The a(4) = 1 through a(8) = 12 partitions:
  (2,1,1)  (2,2,1)    (4,1,1)      (3,2,2)        (3,3,2)
           (3,1,1)    (3,1,1,1)    (3,3,1)        (4,2,2)
           (2,1,1,1)  (2,1,1,1,1)  (5,1,1)        (6,1,1)
                                   (2,2,2,1)      (3,2,2,1)
                                   (3,2,1,1)      (4,2,1,1)
                                   (4,1,1,1)      (5,1,1,1)
                                   (2,2,1,1,1)    (2,2,2,1,1)
                                   (3,1,1,1,1)    (3,2,1,1,1)
                                   (2,1,1,1,1,1)  (4,1,1,1,1)
                                                  (2,2,1,1,1,1)
                                                  (3,1,1,1,1,1)
                                                  (2,1,1,1,1,1,1)

More on set multipartitions: A089259, A116540, A270995, A296119, A318360.
Twice-partitions of this type are counted by A279788.
Interchanging "constant" with "strict" gives A381717, see A381635, A381636, A381991.
Normal multiset partitions of this type are counted by A381718, see A279785.
These partitions are ranked by A381719, zeros of A382080.
For distinct instead of equal block-sums we have A381990, ranked by A381806.
For constant instead of strict blocks we have A381993.
A000041 counts integer partitions, strict A000009.
A050320 counts factorizations into squarefree numbers, see A381078, A381454.
A050326 counts factorizations into distinct squarefree numbers.
A265947 counts refinement-ordered pairs of integer partitions.
A381633 counts set systems with distinct sums, see A381634, A293243.
Cf. A002846, A047966, A279786, A279789, A293511, A299202, A317142, A358914, A381992.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]& /@ sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]& /@ sps[Range[Length[set]]]];
Table[Length[Select[IntegerPartitions[n], Length[Select[mps[#], And@@UnsameQ@@@#&&SameQ@@Total/@#&]]==0&]],{n,0,10}]

A384350 Number of subsets of {1..n} containing at least one element that is a sum of distinct non-elements.

Original entry on oeis.org

0, 0, 0, 1, 4, 13, 33, 81, 183, 402, 856, 1801, 3721, 7646, 15567, 31575

Offset: 0

Gus Wiseman, Jun 05 2025

Conjecture: Also the number of subsets of {1..n} such that it is possible in more than one way to choose a disjoint family of strict integer partitions, one of each element.

			For the set s = {1,5} we have 5 = 2+3, so s is counted under a(5).
The a(0) = 0 through a(5) = 13 subsets:
  .  .  .  {3}  {3}    {3}
                {4}    {4}
                {2,4}  {5}
                {3,4}  {1,5}
                       {2,4}
                       {2,5}
                       {3,4}
                       {3,5}
                       {4,5}
                       {1,4,5}
                       {2,3,5}
                       {2,4,5}
                       {3,4,5}

The complement is counted by A326080, allowing repeats A326083.
For strict partitions of n instead of subsets of {1..n} we have A384318, ranks A384322.
First differences are A384391.
A048767 is the Look-and-Say transform, fixed points A048768, counted by A217605.
A179009 counts maximally refined strict partitions, ranks A383707.
A239455 counts Look-and-Say or section-sum partitions, ranks A351294 or A381432.
A351293 counts non-Look-and-Say or non-section-sum partitions, ranks A351295 or A381433.
A383706 counts ways to choose disjoint strict partitions of prime indices, non-disjoint A357982, non-strict A299200.
Cf. A279375, A279790, A317141, A317142, A383708, A383710, A384317, A384319, A384320, A384321.

nonsets[y_]:=If[Length[y]==0,{},Rest[Subsets[Complement[Range[Max@@y],y]]]];
Table[Length[Select[Subsets[Range[n]],Intersection[#,Total/@nonsets[#]]!={}&]],{n,0,10}]

A317534 Numbers k such that the poset of factorizations of k, ordered by refinement, is not a lattice.

Original entry on oeis.org

24, 32, 40, 48, 54, 56, 60, 64, 72, 80, 84, 88, 90, 96, 104, 108, 112, 120, 126, 128, 132, 135, 136, 140, 144, 150, 152, 156, 160, 162, 168, 176, 180, 184, 189, 192, 198, 200, 204, 208, 216, 220, 224, 228, 232, 234, 240, 243, 248, 250, 252, 256, 260, 264, 270

Offset: 1

Gus Wiseman, Jul 30 2018

nonn

Includes 2^k for all k > 4.

Conjecture: Let S be the set of all numbers whose prime signature is either {1,3}, {5}, or {1,1,2}. Then the sequence consists of all multiples of elements of S. - David A. Corneth, Jul 31 2018.

			In the poset of factorizations of 24, the factorizations (2*2*6) and (2*3*4) have two least-upper bounds, namely (2*12) and (4*6), so this poset is not a lattice.

R. P Stanley, Enumerative Combinatorics Vol. 1, Sec. 3.3.

Wikipedia, Lattice (order)

Cf. A001055, A007716, A025487, A045778, A065036, A162247, A265947, A281113, A317142, A317144, A317145, A317146.

cp's OEIS Frontend

A384390 Heinz numbers of integer partitions with a unique proper way to choose disjoint strict partitions of each part.

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A382079 Number of integer partitions of n that can be partitioned into a set of sets in exactly one way.

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A384319 Number of strict integer partitions of n with exactly two possible ways to choose disjoint strict partitions of each part.

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A384323 Number of integer partitions of n with exactly two possible ways to choose disjoint strict partitions of each part.

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A384349 Heinz numbers of integer partitions with no proper way to choose disjoint strict partitions of each part.

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A381637 Number of multisets that can be obtained by taking the sum of each block of a multiset partition of the prime indices of n into blocks with distinct sums.

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A382460 Number of integer partitions of n that can be partitioned into sets with distinct sums in exactly one way.

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A381994 Number of integer partitions of n that cannot be partitioned into sets with equal sums.

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A384350 Number of subsets of {1..n} containing at least one element that is a sum of distinct non-elements.

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A317534 Numbers k such that the poset of factorizations of k, ordered by refinement, is not a lattice.

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