A118457 -id:A118457 - cp's OEIS frontend

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

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}]

A339351 Irregular triangle read by rows in which row n lists the compositions (ordered partitions) of n into distinct parts in lexicographic order.

Original entry on oeis.org

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

Offset: 1

Ilya Gutkovskiy, Dec 01 2020

			Triangle begins:
[1],
[2],
[1, 2], [2, 1], [3],
[1, 3], [3, 1], [4],
[1, 4], [2, 3], [3, 2], [4, 1], [5],
...

Index entries for sequences related to compositions

Cf. A026793, A066099, A097910 (row lengths), A118457, A228369, A246688, A304797 (row sums), A339178.

Table[Sort[Join @@ Permutations /@ Select[IntegerPartitions[n], UnsameQ @@ # &], OrderedQ[PadRight[{#1, #2}]] &], {n, 8}] // Flatten

A231807 Number of endofunctions on [n] with distinct cardinalities of the nonempty preimages.

Original entry on oeis.org

1, 1, 2, 21, 52, 305, 7836, 24703, 155688, 1034433, 67124260, 235173191, 1728147312, 11309344813, 106962615592, 14055613872945, 55558358852176, 450373499691137, 3156524223157332, 28327606849223119, 307533111218771040, 81782486813477643501

Offset: 0

Alois P. Heinz, Nov 13 2013

nonn

Number of endofunctions f:{1,...,n}-> {1,...,n} such that (1<=i0 and |f^(-1)(j)|>0) implies |f^(-1)(i)| != |f^(-1)(j)|.

			a(3) = 3! * (multinomial(3;3)/2! + multinomial(3;2,1)/1!) = 3+18 = 21: (1,1,1), (2,2,2), (3,3,3), (1,1,2), (1,1,3), (1,2,1), (1,3,1), (2,1,1), (3,1,1), (2,2,1), (2,2,3), (2,1,2), (2,3,2), (1,2,2), (3,2,2), (3,3,1), (3,3,2), (3,1,3), (3,2,3), (1,3,3), (2,3,3).
a(4) = 52: (1,1,1,1), (1,1,1,2), (1,1,1,3), ..., (4,4,4,2), (4,4,4,3), (4,4,4,4).

Alois P. Heinz, Table of n, a(n) for n = 0..637

Column k=1 of A231915.

Cf. A000009, A000312, A231812.

b:= proc(t, i, u) option remember; `if`(t=0, 1, `if`(i<1, 0,
       b(t, i-1, u) +`if`(i>t, 0, b(t-i, i-1, u-1)*u*binomial(t,i))))
    end:
a:= n-> b(n$3):
seq(a(n), n=0..25);

b[t_, i_, u_] := b[t, i, u] = If[t == 0, 1, If[i < 1, 0, b[t, i - 1, u] + If[i > t, 0, b[t - i, i - 1, u - 1] u Binomial[t, i]]]];
a[n_] := b[n, n, n];
a /@ Range[0, 25] (* Jean-François Alcover, Dec 10 2020, after Alois P. Heinz *)

a(n) = n! * Sum_{lambda} multinomial(n;lambda)/(n-|lambda|)!, where lambda ranges over all partitions of n into distinct parts (A118457).

A118459 Lengths of partitions into distinct parts in Mathematica order.

Original entry on oeis.org

0, 1, 1, 1, 2, 1, 2, 1, 2, 2, 1, 2, 2, 3, 1, 2, 2, 2, 3, 1, 2, 2, 2, 3, 3, 1, 2, 2, 2, 3, 2, 3, 3, 1, 2, 2, 2, 3, 2, 3, 3, 3, 4, 1, 2, 2, 2, 3, 2, 3, 2, 3, 3, 3, 4, 1, 2, 2, 2, 3, 2, 3, 2, 3, 3, 3, 3, 4, 3, 4

Offset: 0

Franklin T. Adams-Watters, Apr 28 2006

nonn

Where does this first differ from A118458? - R. J. Mathar, Sep 02 2013

Cf. A118457.

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]
```

A308916 Irregular triangular array: row n shows positions of strict partitions of n among all partitions of n, using Mathematica ordering.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 2, 3, 1, 2, 3, 6, 1, 2, 3, 5, 6, 1, 2, 3, 5, 6, 9, 1, 2, 3, 5, 6, 8, 9, 14, 1, 2, 3, 5, 6, 8, 9, 14, 15, 23, 1, 2, 3, 5, 6, 8, 9, 13, 14, 15, 21, 24, 1, 2, 3, 5, 6, 8, 9, 13, 14, 15, 21, 22, 25, 33, 34, 1, 2, 3, 5, 6, 8, 9, 13, 14, 15

Offset: 1

Clark Kimberling, Jun 30 2019

			1
1
1   2
1   2
1   2   3
1   2   3   6
1   2   3   5   6
1   2   3   5   6   9
1   2   3   5   6   8   9   14
1   2   3   5   6   8   9   14   15   23
Strict partitions of 6: {6}, {5, 1}, {4, 2}, {3, 2, 1}, which occupy positions 1,2,3,6 in the ordering of all partitions of 6: {6}, {5, 1}, {4, 2}, {4, 1, 1}, {3, 3}, {3, 2, 1}, {3, 1, 1, 1}, {2, 2, 2}, {2, 2, 1, 1}, {2, 1, 1, 1, 1}, {1, 1, 1, 1, 1, 1}

Cf. A000009, A000041, A118457, A035399.

p[n_] := IntegerPartitions[n];
d[n_] := Select[p[n], Max[Length /@ Split@#] == 1 &];
t = Table[Flatten[Table[Position[p[n], d[n][[k]]], {k, 1, Length[d[n]]}]], {n, 1, 15}]
Flatten[t]  (* A308916, sequence *)

A339178 Irregular triangle read by rows in which row n lists the compositions (ordered partitions) of n into distinct parts in reverse lexicographic order.

Original entry on oeis.org

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

Offset: 1

Ilya Gutkovskiy, Nov 26 2020

			Triangle begins:
[1],
[2],
[3], [2, 1], [1, 2],
[4], [3, 1], [1, 3],
[5], [4, 1], [3, 2], [2, 3], [1, 4],
...

Index entries for sequences related to compositions

Cf. A026793, A066099, A097910 (row lengths), A118457, A228369, A246688, A304797 (row sums).

Table[Sort[Join @@ Permutations /@ Select[IntegerPartitions[n], UnsameQ @@ # &], OrderedQ[PadRight[{#2, #1}]] &], {n, 8}] // Flatten

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

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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A339351 Irregular triangle read by rows in which row n lists the compositions (ordered partitions) of n into distinct parts in lexicographic order.

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Mathematica

A231807 Number of endofunctions on [n] with distinct cardinalities of the nonempty preimages.

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Maple

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Formula

A118459 Lengths of partitions into distinct parts in Mathematica order.

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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

A308916 Irregular triangular array: row n shows positions of strict partitions of n among all partitions of n, using Mathematica ordering.

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Examples

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Programs

Mathematica

A339178 Irregular triangle read by rows in which row n lists the compositions (ordered partitions) of n into distinct parts in reverse lexicographic order.

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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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