A334441 -id:A334441 - cp's OEIS frontend

A335123 Minimum part of the n-th integer partition in Abramowitz-Stegun order (sum/length/lex); a(0) = 0.

0, 1, 2, 1, 3, 1, 1, 4, 2, 1, 1, 1, 5, 2, 1, 1, 1, 1, 1, 6, 3, 2, 1, 2, 1, 1, 1, 1, 1, 1, 7, 3, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 8, 4, 3, 2, 1, 2, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 9, 4, 3, 2, 1, 3, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 0

Gus Wiseman, May 24 2020

			Triangle begins:
  0
  1
  2 1
  3 1 1
  4 2 1 1 1
  5 2 1 1 1 1 1
  6 3 2 1 2 1 1 1 1 1 1
  7 3 2 1 2 1 1 1 1 1 1 1 1 1 1
  8 4 3 2 1 2 2 1 1 1 2 1 1 1 1 1 1 1 1 1 1 1

OEIS Wiki, Orderings of partitions
Wikiversity, Lexicographic and colexicographic order

Row lengths are A000041.
Partition minima of A334301.
The length of the same partition is A036043.
The Heinz number of the same partition is A334433.
The number of distinct parts in the same partition is A334440.
The maximum of the same partition is A334441.
The version for reversed partitions is A335124.
Reversed partitions in Abramowitz-Stegun (sum/length/lex) order are A036036.
Partitions in (sum/length/revlex) order are A334439.
Cf. A026791, A049085, A103921, A124734, A185974, A193073, A334302, A334433.

Table[If[n==0,{0},Min/@Sort[IntegerPartitions[n]]],{n,0,8}]

a(n) = A055396(A334433(n)).

A344085 Triangle of squarefree numbers first grouped by greatest prime factor, then sorted by omega, then in increasing order, read by rows.

Original entry on oeis.org

1, 2, 3, 6, 5, 10, 15, 30, 7, 14, 21, 35, 42, 70, 105, 210, 11, 22, 33, 55, 77, 66, 110, 154, 165, 231, 385, 330, 462, 770, 1155, 2310, 13, 26, 39, 65, 91, 143, 78, 130, 182, 195, 273, 286, 429, 455, 715, 1001, 390, 546, 858, 910, 1365, 1430, 2002, 2145, 3003, 5005, 2730, 4290, 6006, 10010, 15015, 30030

Offset: 1

Gus Wiseman, May 11 2021

Differs from A339195 in having 77 before 66.

			Triangle begins:
   1
   2
   3   6
   5  10  15  30
   7  14  21  35  42  70 105 210

Row lengths are A000079.
Grouping by greatest prime factor only gives A339195.
Row sums are 1 and A339360.
Cf. A001221, A005117, A005183, A014466, A019565, A187769, A209862.
Partition/composition orderings: A026791, A026792, A026793, A036036, A036037, A048793, A066099, A080577, A112798, A118457, A124734, A162247, A193073, A211992, A228100, A228531, A246688, A272020, A299755, A296774, A304038, A319247, A329631, A334301, A334302, A334439, A334442, A335122, A344086, A344087, A344088, A344089.
Partition/composition applications: A001793, A036043, A049085, A070939, A115623, A124736, A129129, A185974, A238966, A294648, A333483, A333484, A333485, A333486, A334433, A334434, A334435, A334436, A334437, A334438, A334440, A334441, A335123, A335124.

nn=4;
GatherBy[SortBy[Select[Range[Times@@Prime/@Range[nn]],SquareFreeQ[#]&&PrimePi[FactorInteger[#][[-1,1]]]<=nn&],PrimeOmega],FactorInteger[#][[-1,1]]&]

A344090 Flattened tetrangle of strict integer partitions, sorted first by sum, then by length, then lexicographically.

Original entry on oeis.org

1, 2, 3, 2, 1, 4, 3, 1, 5, 3, 2, 4, 1, 6, 4, 2, 5, 1, 3, 2, 1, 7, 4, 3, 5, 2, 6, 1, 4, 2, 1, 8, 5, 3, 6, 2, 7, 1, 4, 3, 1, 5, 2, 1, 9, 5, 4, 6, 3, 7, 2, 8, 1, 4, 3, 2, 5, 3, 1, 6, 2, 1, 10, 6, 4, 7, 3, 8, 2, 9, 1, 5, 3, 2, 5, 4, 1, 6, 3, 1, 7, 2, 1, 4, 3, 2, 1

Offset: 0

Gus Wiseman, May 12 2021

The zeroth row contains only the empty partition.

A tetrangle is a sequence of finite triangles.

			Tetrangle begins:
  0: ()
  1: (1)
  2: (2)
  3: (3)(21)
  4: (4)(31)
  5: (5)(32)(41)
  6: (6)(42)(51)(321)
  7: (7)(43)(52)(61)(421)
  8: (8)(53)(62)(71)(431)(521)
  9: (9)(54)(63)(72)(81)(432)(531)(621)

Wikiversity, Lexicographic and colexicographic order

Starting with reversed partitions gives A026793.
The version for compositions is A124734.
Showing partitions as Heinz numbers gives A246867.
The non-strict version is A334301 (reversed: A036036).
Ignoring length gives A344086 (reversed: A246688).
Same as A344089 with partitions reversed.
The version for revlex instead of lex is A344092.
A026791 reads off lexicographically ordered reversed partitions.
A080577 reads off reverse-lexicographically ordered partitions.
A112798 reads off reversed partitions by Heinz number.
A296150 reads off partitions by Heinz number.
Cf. A036037, A036043, A103921, A185974, A193073, A211992, A296774, A334302, A334433, A334435, A334438, A334439, A334440, A334441, A334442, A344091.

Table[Sort[Select[IntegerPartitions[n],UnsameQ@@#&]],{n,0,10}]

A344087 Flattened tetrangle of strict integer partitions sorted first by sum, then colexicographically.

Original entry on oeis.org

1, 2, 2, 1, 3, 3, 1, 4, 4, 1, 3, 2, 5, 3, 2, 1, 5, 1, 4, 2, 6, 4, 2, 1, 6, 1, 5, 2, 4, 3, 7, 5, 2, 1, 4, 3, 1, 7, 1, 6, 2, 5, 3, 8, 6, 2, 1, 5, 3, 1, 8, 1, 4, 3, 2, 7, 2, 6, 3, 5, 4, 9, 4, 3, 2, 1, 7, 2, 1, 6, 3, 1, 5, 4, 1, 9, 1, 5, 3, 2, 8, 2, 7, 3, 6, 4, 10

Offset: 0

Gus Wiseman, May 11 2021

The zeroth row contains only the empty partition.

A tetrangle is a sequence of finite triangles.

			Tetrangle begins:
  0: ()
  1: (1)
  2: (2)
  3: (21)(3)
  4: (31)(4)
  5: (41)(32)(5)
  6: (321)(51)(42)(6)
  7: (421)(61)(52)(43)(7)
  8: (521)(431)(71)(62)(53)(8)
  9: (621)(531)(81)(432)(72)(63)(54)(9)

Wikiversity, Lexicographic and colexicographic order

Positions of first appearances are A015724.
Triangle sums are A066189.
Taking revlex instead of colex gives A118457.
The not necessarily strict version is A211992.
Taking lex instead of colex gives A344086.
A026793 gives reversed strict partitions in A-S order (sum/length/lex).
A319247 sorts strict partitions by Heinz number.
A329631 sorts reversed strict partitions by Heinz number.
A344090 gives strict partitions in A-S order (sum/length/lex).
Cf. A005117, A014466, A209862.
Partition/composition orderings: A026791, A026792, A036036, A036037, A048793, A066099, A080576, A080577, A112798, A124734, A162247, A193073, A211992, A228100, A228351, A228531, A246688, A272020, A299755, A296774, A304038, A319247, A334301, A334302, A334439, A334442, A335122, A339351, A344085, A344086, A344088, A344089, A344091.
Partition/composition applications: A001793, A005183, A036043, A049085, A070939, A115623, A124736, A129129, A185974, A238966, A246867, A294648, A333483, A333484, A333485, A333486, A334433, A334434, A334435, A334436, A334437, A334438, A334440, A334441, A335123, A335124, A339195.

colex[f_,c_]:=OrderedQ[PadRight[{Reverse[f],Reverse[c]}]];
Table[Sort[Select[IntegerPartitions[n],UnsameQ@@#&],colex],{n,0,10}]

A344088 Flattened tetrangle of reversed strict integer partitions sorted first by sum, then colexicographically.

Original entry on oeis.org

1, 2, 1, 2, 3, 1, 3, 4, 2, 3, 1, 4, 5, 1, 2, 3, 2, 4, 1, 5, 6, 1, 2, 4, 3, 4, 2, 5, 1, 6, 7, 1, 3, 4, 1, 2, 5, 3, 5, 2, 6, 1, 7, 8, 2, 3, 4, 1, 3, 5, 4, 5, 1, 2, 6, 3, 6, 2, 7, 1, 8, 9, 1, 2, 3, 4, 2, 3, 5, 1, 4, 5, 1, 3, 6, 4, 6, 1, 2, 7, 3, 7, 2, 8, 1, 9, 10

Offset: 0

Gus Wiseman, May 12 2021

The zeroth row contains only the empty partition.

A tetrangle is a sequence of finite triangles.

			Tetrangle begins:
  0: ()
  1: (1)
  2: (2)
  3: (12)(3)
  4: (13)(4)
  5: (23)(14)(5)
  6: (123)(24)(15)(6)
  7: (124)(34)(25)(16)(7)
  8: (134)(125)(35)(26)(17)(8)
  9: (234)(135)(45)(126)(36)(27)(18)(9)

Wikiversity, Lexicographic and colexicographic order

Positions of first appearances are A015724.
Triangle sums are A066189.
The non-strict version is A080576.
Taking lex instead of colex gives A246688 (non-reversed: A344086).
The non-reversed version is A344087.
Taking revlex instead of colex gives A344089 (non-reversed: A118457).
A026793 gives reversed strict partitions in A-S order (sum/length/lex).
A319247 sorts strict partitions by Heinz number.
A329631 sorts reversed strict partitions by Heinz number.
A344090 gives strict partitions in A-S order (sum/length/lex).
Cf. A005117, A014466, A209862.
Partition/composition orderings: A026791, A026792, A036036, A036037, A048793, A066099, A080577, A112798, A124734, A162247, A193073, A211992, A228100, A228351, A228531, A272020, A299755, A296774, A304038, A334301, A334302, A334439, A334442, A335122, A339351, A344085, A344086, A344091.
Partition/composition applications: A001793, A005183, A036043, A049085, A070939, A115623, A124736, A129129, A185974, A238966, A246867, A294648, A333483, A333484, A333485, A333486, A334433, A334434, A334435, A334436, A334437, A334438, A334440, A334441, A335123, A335124, A339195.

colex[f_,c_]:=OrderedQ[PadRight[{Reverse[f],Reverse[c]}]];
Table[Sort[Reverse/@Select[IntegerPartitions[n],UnsameQ@@#&],colex],{n,0,10}]

A344091 Flattened tetrangle of all finite multisets of positive integers sorted first by sum, then by length, then colexicographically.

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, May 12 2021

First differs from A334302 for partitions of 9.
The zeroth row contains only the empty partition.
A tetrangle is a sequence of finite triangles.

			Tetrangle begins:
  0: ()
  1: (1)
  2: (2)(11)
  3: (3)(12)(111)
  4: (4)(22)(13)(112)(1111)
  5: (5)(23)(14)(122)(113)(1112)(11111)
  6: (6)(33)(24)(15)(222)(123)(114)(1122)(1113)(11112)(111111)

Wikiversity, Lexicographic and colexicographic order

The version for lex instead of colex is A036036.
Starting with reversed partitions gives A036037.
Ignoring length gives A211992 (reversed: A080576).
Same as A334301 with partitions reversed.
The version for revlex instead of colex is A334302.
The Heinz numbers of these partitions are A334433.
The strict case is A344089.
A026791 reads off lexicographically ordered reversed partitions.
A080577 reads off reverse-lexicographically ordered partitions.
A112798 reads off reversed partitions by Heinz number.
A193073 reads off lexicographically ordered partitions.
A296150 reads off partitions by Heinz number.
Cf. A000041, A026793, A124734, A185974, A228531, A246688, A296774, A334433, A334435, A334438, A334439, A334441, A334442, A344086, A344090.

Table[Reverse/@Sort[IntegerPartitions[n]],{n,0,9}]

A344084 Concatenated list of all finite nonempty sets of positive integers sorted first by maximum, then by length, and finally lexicographically.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, May 11 2021

			The sets are the columns below:
  1 2 1 3 1 2 1 4 1 2 3 1 1 2 1 5 1 2 3 4 1 1 1 2 2 3 1
      2   3 3 2   4 4 4 2 3 3 2   5 5 5 5 2 3 4 3 4 4 2
              3         4 4 4 3           5 5 5 5 5 5 3
                              4                       5
As a tetrangle, the first four triangles are:
  {1}
  {2},{1,2}
  {3},{1,3},{2,3},{1,2,3}
  {4},{1,4},{2,4},{3,4},{1,2,4},{1,3,4},{2,3,4},{1,2,3,4}

Wikiversity, Lexicographic and colexicographic order

Triangle lengths are A000079.
Triangle sums are A001793.
Positions of first appearances are A005183.
Set maxima are A070939.
Set lengths are A124736.
Cf. A005117, A014466, A209862.
Partition/composition orderings: A026791, A026792, A026793, A036036, A036037, A048793, A066099, A080577, A112798, A118457, A124734, A162247, A193073, A211992, A228100, A228531, A246688, A272020, A296774, A299755, A304038, A319247, A329631, A334301, A334302, A334439, A334442, A335122, A344085, A344086, A344087, A344088, A344089.
Partition/composition applications: A036043, A049085, A115623, A129129, A185974, A238966, A294648, A333483, A333484, A333485, A333486, A334433, A334434, A334435, A334436, A334437, A334438, A334440, A334441, A335123, A335124, A339195.

Mathematica
```
SortBy[Rest[Subsets[Range[5]]],Last]
```

A344092 Flattened tetrangle of strict integer partitions, sorted first by sum, then by length, and finally reverse-lexicographically.

Original entry on oeis.org

1, 2, 3, 2, 1, 4, 3, 1, 5, 4, 1, 3, 2, 6, 5, 1, 4, 2, 3, 2, 1, 7, 6, 1, 5, 2, 4, 3, 4, 2, 1, 8, 7, 1, 6, 2, 5, 3, 5, 2, 1, 4, 3, 1, 9, 8, 1, 7, 2, 6, 3, 5, 4, 6, 2, 1, 5, 3, 1, 4, 3, 2, 10, 9, 1, 8, 2, 7, 3, 6, 4, 7, 2, 1, 6, 3, 1, 5, 4, 1, 5, 3, 2, 4, 3, 2, 1

Offset: 0

Gus Wiseman, May 14 2021

First differs from A118457 at a(53) = 4, A118457(53) = 2.
The zeroth row contains only the empty partition.
A tetrangle is a sequence of finite triangles.

			Tetrangle begins:
   0: ()
   1: (1)
   2: (2)
   3: (3)(21)
   4: (4)(31)
   5: (5)(41)(32)
   6: (6)(51)(42)(321)
   7: (7)(61)(52)(43)(421)
   8: (8)(71)(62)(53)(521)(431)
   9: (9)(81)(72)(63)(54)(621)(531)(432)

Wikiversity, Lexicographic and colexicographic order

Same as A026793 with rows reversed.
Ignoring length gives A118457.
The non-strict version is A334439 (reversed: A036036/A334302).
The version for lex instead of revlex is A344090.
A026791 reads off lexicographically ordered reversed partitions.
A080577 reads off reverse-lexicographically ordered partitions.
A112798 reads off reversed partitions by Heinz number.
A193073 reads off lexicographically ordered partitions.
A296150 reads off partitions by Heinz number.
Cf. A036037, A036043, A103921, A124734, A185974, A211992, A296774, A334301, A334433, A334435, A334438, A334441.

revlensort[f_,c_]:=If[Length[f]!=Length[c],Length[f]

cp's OEIS Frontend

A335123 Minimum part of the n-th integer partition in Abramowitz-Stegun order (sum/length/lex); a(0) = 0.

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A344085 Triangle of squarefree numbers first grouped by greatest prime factor, then sorted by omega, then in increasing order, read by rows.

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A344090 Flattened tetrangle of strict integer partitions, sorted first by sum, then by length, then lexicographically.

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A344087 Flattened tetrangle of strict integer partitions sorted first by sum, then colexicographically.

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A344088 Flattened tetrangle of reversed strict integer partitions sorted first by sum, then colexicographically.

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A344091 Flattened tetrangle of all finite multisets of positive integers sorted first by sum, then by length, then colexicographically.

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Mathematica

A344084 Concatenated list of all finite nonempty sets of positive integers sorted first by maximum, then by length, and finally lexicographically.

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Mathematica

A344092 Flattened tetrangle of strict integer partitions, sorted first by sum, then by length, and finally reverse-lexicographically.

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