A026793 -id:A026793 - cp's OEIS frontend

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

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

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}]

A118458 Lengths of partitions into distinct parts in Abramowitz and Stegun 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

			The triangle starts in row n=0 as:
0;
1;
1;
1,2;
1,2;
1,2,2;
1,2,2,3;
The 6th row is 1 (length of 6), 2 (length of 1+5), 2 (length of 2+4), and 3 (length of 1+2+3).

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

Cf. A026793, A118458

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]
```

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]

A379756 a(n) is the number of subsets of S(n) that sum to A023196(n), where S(n) is the set of the proper divisors (or aliquot parts) of A023196(n).

Original entry on oeis.org

1, 2, 2, 1, 5, 1, 3, 7, 3, 2, 10, 3, 2, 34, 2, 0, 31, 1, 6, 25, 1, 23, 21, 2, 1, 1, 20, 4, 1, 279, 13, 15, 1, 15, 116, 9, 11, 12, 4, 197, 1, 2, 755, 1, 42, 2, 9, 12, 6, 2, 151, 169, 7, 1, 9, 8, 6, 2190, 1, 516, 1, 6, 121, 130, 1, 6, 119, 1, 469, 4, 446, 1, 4, 6

Offset: 1

Felix Huber, Feb 07 2025

nonn

This sequence is A065205 without the terms A065205(k) where k > sigma(k)/2.

			a(8) = 7 because exactly the 7 subsets {6, 12, 18}, {3, 6, 9, 18}, {2, 4, 12, 18}, {2, 3, 4, 9, 18}, {2, 3, 4, 6, 9, 12}, {1, 2, 6, 9, 18}, {1, 2, 3, 12, 18} of S(8) = {1, 2, 3, 4, 6, 9, 12, 18} sum to A023196(8) = 36.
a(16) = 0 because no subset of S(16) = {1, 2, 5, 7, 10, 14, 35} sums to A023196(16) = 70 (weird number).

Felix Huber, Table of n, a(n) for n = 1..10000
Felix Huber, Maple program to calculate the distinct subsets

Cf. A000203, A000396, A001065, A005101, A023196, A026793, A027750, A033630, A033882, A064771, A065205.

with(NumberTheory):
A023196:=proc(n)
    local a;
    option remember;
    if n=1 then
        6
    else
        for a from procname(n-1)+1 do
            if sigma(a)>=2*a then
                return a
            fi
        od
    fi;
end proc;
A379756:=proc(n)
    local b,d,l;
    d:=sigma(A023196(n))-2*A023196(n);
    l:= [select(x->x<=d,Divisors(A023196(n)))[]];
    b:= proc(m,i)
        option remember;
        `if`(m=0,1,`if`(i<1,0,b(m,i-1)+`if`(l[i]>m,0,b(m-l[i],i-1))))
    end proc;
    forget(b);
    b(d,nops(l))
end proc;
seq(A379756(n),n=1..74);

Iff a(k) = 0, A023196(k) is a weird number (A006037).
Iff a(k) = 1, A023196(k) is a term of A064771.
a(A000396(k)) = 1 (A000396: perfect numbers).

cp's OEIS Frontend

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A118458 Lengths of partitions into distinct parts in Abramowitz and Stegun order.

Views

Author

Keywords

Examples

Links

Crossrefs

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

Views

Author

Keywords

Examples

Links

Crossrefs

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.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A379756 a(n) is the number of subsets of S(n) that sum to A023196(n), where S(n) is the set of the proper divisors (or aliquot parts) of A023196(n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Formula