A124734 -id:A124734 - 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)).

A335124 Minimum part of the n-th reversed integer partition in Abramowitz-Stegun order; a(0) = 0.

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, May 24 2020

The ordering of reversed partitions is first by sum, then by length, and finally lexicographically. The version for non-reversed partitions is A335123.

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

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

Row lengths are A000041.
Partition minima of A036036.
The length of the same partition is A036043.
The maximum of the same partition is A049085.
The number of distinct parts in the same partition is A103921.
The Heinz number of the same partition is A185974.
The version for non-reversed partitions is A335123.
Lexicographically ordered reversed partitions are A026791.
Partitions in (sum/length/colex) order are A036037.
Partitions in opposite Abramowitz-Stegun (sum/length/revlex) order are A334439.
Cf. A124734, A193073, A334301, A334302, A334433, A334435.

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

a(n) = A055396(A185974(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]]&]

A296656 Triangle whose n-th row is the concatenated sequence of all Lyndon compositions of n in reverse-lexicographic order.

Original entry on oeis.org

1, 2, 3, 1, 2, 4, 1, 3, 1, 1, 2, 5, 2, 3, 1, 4, 1, 2, 2, 1, 1, 3, 1, 1, 1, 2, 6, 2, 4, 1, 5, 1, 3, 2, 1, 2, 3, 1, 1, 4, 1, 1, 2, 2, 1, 1, 1, 3, 1, 1, 1, 1, 2, 7, 3, 4, 2, 5, 2, 2, 3, 1, 6, 1, 4, 2, 1, 3, 3, 1, 2, 4, 1, 2, 2, 2, 1, 2, 1, 3, 1, 1, 5, 1, 1, 3, 2

Offset: 1

Gus Wiseman, Dec 18 2017

			Triangle of Lyndon compositions begins:
(1),
(2),
(3),(12),
(4),(13),(112),
(5),(23),(14),(122),(113),(1112),
(6),(24),(15),(132),(123),(114),(1122),(1113),(11112),
(7),(34),(25),(223),(16),(142),(133),(124),(1222),(1213),(115),(1132),(1123),(11212),(1114),(11122),(11113),(111112).

Cf. A000740, A001037, A001045, A008965, A059966, A060223, A066099, A101211, A102659, A124734, A185700, A228369, A281013, A294859, A296302, A296373.

LyndonQ[q_]:=Array[OrderedQ[{q,RotateRight[q,#]}]&,Length[q]-1,1,And]&&Array[RotateRight[q,#]&,Length[q],1,UnsameQ];
Table[Sort[Select[Join@@Permutations/@IntegerPartitions[n],LyndonQ],OrderedQ[PadRight[{#2,#1}]]&],{n,7}]

Row n is a concatenation of A059966(n) Lyndon words with total length A000740(n).

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}]

A337243 Compositions, sorted by increasing sum, increasing length, and increasing colexicographical order.

Original entry on oeis.org

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

Offset: 1

Lorenzo Sauras Altuzarra, Aug 21 2020

			The first 5 rows are:
(1),
(2), (1, 1),
(3), (2, 1), (1, 2), (1, 1, 1),
(4), (3, 1), (2, 2), (1, 3), (2, 1, 1), (1, 2, 1), (1, 1, 2), (1, 1, 1, 1),
(5), (4, 1), (3, 2), (2, 3), (1, 4), (3, 1, 1), (2, 2, 1), (1, 3, 1), (2, 1, 2), (1, 2, 2), (1, 1, 3), (2, 1, 1, 1), (1, 2, 1, 1), (1, 1, 2, 1), (1, 1, 1, 2), (1, 1, 1, 1, 1).

Cf. A124734 (increasing length, then lexicographic).
Cf. A296774 (increasing length, then reverse lexicographic).
Cf. A337259 (increasing length, then reverse colexicographic).
Cf. A296773 (decreasing length, then lexicographic).
Cf. A296772 (decreasing length, then reverse lexicographic).
Cf. A337260 (decreasing length, then colexicographic).
Cf. A108244 (decreasing length, then reverse colexicographic).
Cf. A228369 (lexicographic).
Cf. A066099 (reverse lexicographic).
Cf. A228525 (colexicographic).
Cf. A228351 (reverse colexicographic).

List := proc(n)
   local i, j, k, L:
   L := []:
   for i from 1 to n do
      for j from 1 to i do
         L := [op(L), op(combinat:-composition(i, j))]:
      od:
   od:
   for k from 1 to numelems(L) do L[k] := ListTools:-Reverse(L[k]): od:
   L:
end:

A337259 Compositions, sorted by increasing sum, increasing length and decreasing colexicographical order.

Original entry on oeis.org

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

Offset: 1

Lorenzo Sauras Altuzarra, Aug 21 2020

			The first 5 rows are:
(1),
(2), (1, 1),
(3), (1, 2), (2, 1), (1, 1, 1),
(4), (1, 3), (2, 2), (3, 1), (1, 1, 2), (1, 2, 1), (2, 1, 1), (1, 1, 1, 1),
(5), (1, 4), (2, 3), (3, 2), (4, 1), (1, 1, 3), (1, 2, 2), (2, 1, 2), (1, 3, 1), (2, 2, 1), (3, 1, 1), (1, 1, 1, 2), (1, 1, 2, 1), (1, 2, 1, 1), (2, 1, 1, 1), (1, 1, 1, 1, 1).

Cf. A124734 (increasing length, then lexicographic).
Cf. A296774 (increasing length, then reverse lexicographic).
Cf. A337243 (increasing length, then colexicographic).
Cf. A296773 (decreasing length, then lexicographic).
Cf. A296772 (decreasing length, then reverse lexicographic).
Cf. A337260 (decreasing length, then colexicographic).
Cf. A108244 (decreasing length, then reverse colexicographic).
Cf. A228369 (lexicographic).
Cf. A066099 (reverse lexicographic).
Cf. A228525 (colexicographic).
Cf. A228351 (reverse colexicographic).

List := proc(n)
   local i, j, k, L:
   L := []:
   for i from 1 to n do
      for j from 1 to i do
         L := [op(L), op(ListTools:-Reverse([op(combinat:-composition(i, j))]))]:
      od:
   od:
   for k from 1 to numelems(L) do L[k] := ListTools:-Reverse(L[k]): od:
   L:
end:

A337260 Compositions, sorted by increasing sum, decreasing length and increasing colexicographical order.

Original entry on oeis.org

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

Offset: 1

Lorenzo Sauras Altuzarra, Aug 21 2020

			The first 5 rows are:
(1),
(1, 1), (2),
(1, 1, 1), (2, 1), (1, 2), (3),
(1, 1, 1, 1), (2, 1, 1), (1, 2, 1), (1, 1, 2), (3, 1), (2, 2), (1, 3), (4),
(1, 1, 1, 1, 1), (2, 1, 1, 1), (1, 2, 1, 1), (1, 1, 2, 1), (1, 1, 1, 2), (3, 1, 1), (2, 2, 1), (1, 3, 1), (2, 1, 2), (1, 2, 2), (1, 1, 3), (4, 1), (3, 2), (2, 3), (1, 4), (5).

Cf. A124734 (increasing length, then lexicographic).
Cf. A296774 (increasing length, then reverse lexicographic).
Cf. A337243 (increasing length, then colexicographic).
Cf. A337259 (increasing length, then reverse colexicographic).
Cf. A296773 (decreasing length, then lexicographic).
Cf. A296772 (decreasing length, then reverse lexicographic).
Cf. A108244 (decreasing length, then reverse colexicographic).
Cf. A228369 (lexicographic).
Cf. A066099 (reverse lexicographic).
Cf. A228525 (colexicographic).
Cf. A228351 (reverse colexicographic).

List := proc(n)
   local i, j, k, L:
   L := []:
   for i from 1 to n do
      for j from 1 to i do
         L := [op(L), op(combinat:-composition(i, i-j+1))]:
      od:
   od:
   for k from 1 to numelems(L) do L[k] := ListTools:-Reverse(L[k]): od:
   L:
end:

cp's OEIS Frontend

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A335124 Minimum part of the n-th reversed integer partition in Abramowitz-Stegun order; a(0) = 0.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A296656 Triangle whose n-th row is the concatenated sequence of all Lyndon compositions of n in reverse-lexicographic order.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A337243 Compositions, sorted by increasing sum, increasing length, and increasing colexicographical order.

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

A337259 Compositions, sorted by increasing sum, increasing length and decreasing colexicographical order.

Views