A065120 -id:A065120 - cp's OEIS frontend

A123639 Consider the 2^n compositions of n and count only those ending in an even part.

0, 1, 2, 6, 18, 61, 224, 890, 3784, 17113, 81950, 414230, 2204110, 12314109, 72049548, 440379770, 2805266692, 18584809833, 127812870474, 910990458022, 6719535098378, 51223251471453, 403044829472760, 3269538955148698, 27314067026782976, 234749040898160153

Offset: 1

Alford Arnold, Oct 04 2006

nonn

Compositions ending in an even part yield sequence 0 1 2 6 18 ... (this sequence). and A123638(n)+a(n) = A047970(n). Ending parity of compositions can be detected using mod(A065120,2)

			4
31 32 33
211 221 222
1111
Consider the above multisets- permute and note the parity of the ending part of each of the 14 compositions.
4
31 13 32 23 33
211 121 112 221 212 122 222
1111
4 is even
31 13 23 and 33 are odd
32 is even
etc
there are 1+1+4+0 even compositions therefore a(4)=6.

Alois P. Heinz, Table of n, a(n) for n = 1..225

Cf. A001045, A047970, A065120, A123638, A123640, A123641.

g:= proc(b,t,l,m) option remember; if t=0 then b*(1-l) else add (g(b, t-1, irem(k, 2), m), k=1..m-1) +g(1, t-1, irem(m, 2), m) fi end: a:= n-> add (g(0, k, 0, n+1-k), k=1..n): seq (a(n), n=1..30); # Alois P. Heinz, Nov 06 2009

g[b_, t_, l_, m_] := g[b, t, l, m] = If[ t == 0 , b*(1-l), Sum[g[b, t-1, Mod[k, 2], m], {k, 1, m-1}] + g[1, t-1, Mod[m, 2], m]]; a[n_] := Sum[g[0, k, 0, n+1-k], {k, 1, n}]; Table[a[n], {n, 1, 30}] (* Jean-François Alcover, Nov 04 2013, translated from Alois P. Heinz's Maple program *)

More terms from Alois P. Heinz, Nov 06 2009

A228349 Triangle read by rows: T(j,k) is the k-th part in nondecreasing order of the j-th region of the set of compositions (ordered partitions) of n in colexicographic order, if 1<=j<=2^(n-1) and 1<=k<=A006519(j).

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Aug 26 2013

Triangle read by rows in which row n lists the A006519(n) elements of the row A001511(n) of triangle A090996, n >= 1.

The equivalent sequence for partitions is A220482.

			----------------------------------------------------------
.             Diagram                Triangle
Compositions    of            of compositions (rows)
of 5          regions          and regions (columns)
----------------------------------------------------------
.            _ _ _ _ _
5           |_        |                                 5
1+4         |_|_      |                               1 4
2+3         |_  |     |                             2   3
1+1+3       |_|_|_    |                           1 1   3
3+2         |_    |   |                         3       2
1+2+2       |_|_  |   |                       1 2       2
2+1+2       |_  | |   |                     2   1       2
1+1+1+2     |_|_|_|_  |                   1 1   1       2
4+1         |_      | |                 4               1
1+3+1       |_|_    | |               1 3               1
2+2+1       |_  |   | |             2   2               1
1+1+2+1     |_|_|_  | |           1 1   2               1
3+1+1       |_    | | |         3       1               1
1+2+1+1     |_|_  | | |       1 2       1               1
2+1+1+1     |_  | | | |     2   1       1               1
1+1+1+1+1   |_|_|_|_|_|   1 1   1       1               1
.
Written as an irregular triangle in which row n lists the parts of the n-th region the sequence begins:
1;
1,2;
1;
1,1,2,3;
1;
1,2;
1;
1,1,1,1,2,2,3,4;
1;
1,2;
1;
1,1,2,3;
1;
1,2;
1;
1,1,1,1,1,1,1,1,2,2,2,2,3,3,4,5;
...
Alternative interpretation of this sequence:
Triangle read by rows in which row r lists the regions of the last section of the set of compositions of r:
[1];
[1,2];
[1],[1,1,2,3];
[1],[1,2],[1],[1,1,1,1,2,2,3,4];
[1],[1,2],[1],[1,1,2,3],[1],[1,2],[1],[1,1,1,1,1,1,1,1,2,2,2,2,3,3,4,5];

Michael De Vlieger, Table of n, a(n) for n = 1..13312 (rows 1 <= n <= 2^11 = 2048).

Main triangle: Right border gives A001511. Row j has length A006519(j). Row sums give A038712.

Cf. A001787, A001792, A011782, A029837, A045623, A065120, A070939, A090996, A186114, A187816, A187818, A206437, A220482, A228347, A228348, A228350, A228351, A228366, A228367, A228370, A228371, A228525, A228526.

Table[Map[Length@ TakeWhile[IntegerDigits[#, 2], # == 1 &] &, Range[2^(# - 1), 2^# - 1]] &@ IntegerExponent[2 n, 2], {n, 32}] // Flatten (* Michael De Vlieger, May 23 2017 *)

A357180 First run-length of the n-th composition in standard order.

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 1, 3, 1, 1, 2, 1, 1, 1, 2, 4, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 2, 3, 5, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 4, 6, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 2

Offset: 0

Gus Wiseman, Sep 24 2022

nonn

A composition of n is a finite sequence of positive integers summing to n. The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

			Composition 87 in standard order is (2,2,1,1,1), so a(87) = 2.

Gus Wiseman, Statistics, classes, and transformations of standard compositions

See link for sequences related to standard compositions.
For parts instead of run-lengths we have A065120, last A001511.
The version for Heinz numbers of partitions is A067029, last A071178.
This is the first part of row n of A333769.
For minimal instead of first we have A357138, maximal A357137.
The last instead of first run-length is A357181.
A051903 gives maximal part in prime signature.
A061395 gives maximal prime index.
A124767 counts runs in standard compositions.
A286470 gives maximal difference of prime indices.
A333766 gives maximal part of standard compositions, minimal A333768.
A353847 ranks run-sums of standard compositions.
Cf. A000120, A003754, A029931, A070939, A329395, A356841, A356844, A357134, A357135, A357136.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Table[If[n==0,0,First[Length/@Split[stc[n]]]],{n,0,100}]

A375139 Numbers k such that the leaders of strictly increasing runs in the k-th composition in standard order are not weakly decreasing.

Original entry on oeis.org

26, 50, 53, 58, 90, 98, 100, 101, 106, 107, 114, 117, 122, 154, 164, 178, 181, 186, 194, 196, 197, 201, 202, 203, 210, 212, 213, 214, 215, 218, 226, 228, 229, 234, 235, 242, 245, 250, 282, 306, 309, 314, 324, 329, 346, 354, 356, 357, 362, 363, 370, 373, 378

Offset: 1

Gus Wiseman, Aug 12 2024

nonn

The leaders of strictly increasing runs in a sequence are obtained by splitting it into maximal strictly increasing subsequences and taking the first term of each.

The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

			The terms together with corresponding compositions begin:
   26: (1,2,2)
   50: (1,3,2)
   53: (1,2,2,1)
   58: (1,1,2,2)
   90: (2,1,2,2)
   98: (1,4,2)
  100: (1,3,3)
  101: (1,3,2,1)
  106: (1,2,2,2)
  107: (1,2,2,1,1)
  114: (1,1,3,2)
  117: (1,1,2,2,1)
  122: (1,1,1,2,2)
  154: (3,1,2,2)
  164: (2,3,3)
  178: (2,1,3,2)
  181: (2,1,2,2,1)
  186: (2,1,1,2,2)

Gus Wiseman, Sequences counting and ranking compositions by their leaders (for six types of runs).

For leaders of identical runs we have A335485.
Ranked by positions of non-weakly decreasing rows in A374683.
For identical leaders we have A374685, counted by A374686.
The complement is counted by A374697.
For distinct leaders we have A374698, counted by A374687.
Compositions of this type are counted by A375135.
Weakly increasing leaders: A375137, counts A374636, complement A189076.
Interchanging weak/strict: A375295, counted by A375140, complement A188920.
A003242 counts anti-run compositions, ranks A333489.
A374700 counts compositions by sum of leaders of strictly increasing runs.
All of the following pertain to compositions in standard order:
- Length is A000120.
- Sum is A029837(n+1).
- Leader is A065120.
- Parts are listed by A066099.
- Strict compositions are A233564.
- Run-length transform is A333627, sum A070939.
- Run-compression transform is A373948, sum A373953, excess A373954.
- Run-count: A124766, A124765, A124768, A124769, A333381, A124767.
- Run-leaders: A374629, A374740, A374683, A374757, A374515, A374251.
- Run-rank: A375123, A375124, A375125, A375126, A375127.
Cf. A000009, A261982, A333213, A374520, A374630, A374699, A374768.

stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,100],!GreaterEqual@@First/@Split[stc[#],Less]&]

A123638 Consider the 2^n compositions of n and count only those ending in an odd part with row sum A001045.

Original entry on oeis.org

1, 1, 3, 8, 25, 83, 299, 1158, 4813, 21373, 100955, 504916, 2662761, 14754311, 85643459, 519493938, 3285790317, 21628225041, 147887079907, 1048634836288, 7698589399833, 58432476430139, 457901993065915, 3700291495531166

Offset: 1

Alford Arnold, Oct 04 2006

nonn

Compositions ending in an even part yield sequence 0 1 2 6 18 ... A123639. and a(n)+A123639(n) = A047970(n). Ending parity of compositions can be detected using mod(A065120,2)

			4
31 32 33
211 221 222
1111
Consider the above multisets: permute and note the parity of the ending part of each of the 14 compositions.
4
31 13 32 23 33
211 121 112 221 212 122 222
1111
4 is even
31 13 23 and 33 are odd
32 is even
etc
there are 0 + 4 + 3 + 1 = 8 odd compositions therefore a(4)=8.

Cf. A001045 A047970 A065120 A123639 A123640 A123641.

g:= proc(b,t,l,m) option remember; if t=0 then b*l else add (g(b, t-1, irem(k, 2), m), k=1..m-1) +g(1, t-1, irem(m, 2), m) fi end: a:= n-> add (g(0, k, 0, n+1-k), k=1..n): seq (a(n), n=1..30);

g[b_, t_, l_, m_] := g[b, t, l, m] = If[t == 0 , b*l , Sum[g[b, t-1, Mod[k, 2], m], {k, 1, m-1}] + g[1, t-1, Mod[m, 2], m]]; a[n_] := Sum[g[0, k, 0, n+1-k], {k, 1, n}]; Table[a[n], {n, 1, 30}] (* Jean-François Alcover, Nov 04 2013, translated from Alois P. Heinz's Maple program *)

Offset corrected, Maple program and more terms added by Alois P. Heinz, Nov 06 2009

A228347 Triangle of regions and compositions of the positive integers (see Comments lines for definition).

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Aug 26 2013

Triangle read by rows in which row n lists A129760(n) zeros followed by the A006519(n) elements of the row A001511(n) of triangle A090996, n >= 1.

The equivalent sequence for partitions is A186114.

			----------------------------------------------------------
.             Diagram                Triangle
Compositions    of            of compositions (rows)
of 5          regions          and regions (columns)
----------------------------------------------------------
.            _ _ _ _ _
5           |_        |                                 5
1+4         |_|_      |                               1 4
2+3         |_  |     |                             2 0 3
1+1+3       |_|_|_    |                           1 1 0 3
3+2         |_    |   |                         3 0 0 0 2
1+2+2       |_|_  |   |                       1 2 0 0 0 2
2+1+2       |_  | |   |                     2 0 1 0 0 0 2
1+1+1+2     |_|_|_|_  |                   1 1 0 1 0 0 0 2
4+1         |_      | |                 4 0 0 0 0 0 0 0 1
1+3+1       |_|_    | |               1 3 0 0 0 0 0 0 0 1
2+2+1       |_  |   | |             2 0 2 0 0 0 0 0 0 0 1
1+1+2+1     |_|_|_  | |           1 1 0 2 0 0 0 0 0 0 0 1
3+1+1       |_    | | |         3 0 0 0 1 0 0 0 0 0 0 0 1
1+2+1+1     |_|_  | | |       1 2 0 0 0 1 0 0 0 0 0 0 0 1
2+1+1+1     |_  | | | |     2 0 1 0 0 0 1 0 0 0 0 0 0 0 1
1+1+1+1+1   |_|_|_|_|_|   1 1 0 1 0 0 0 1 0 0 0 0 0 0 0 1
.
For the positive integer k consider the first 2^(k-1) rows of triangle, as shown below. The positive terms of the n-th row are the parts of the n-th region of the diagram of regions of the set of compositions of k. The positive terms of the n-th column are the parts of the n-th composition of k, with compositions in colexicographic order.
Triangle begins:
1;
1,2;
0,0,1;
1,1,2,3;
0,0,0,0,1;
0,0,0,0,1,2;
0,0,0,0,0,0,1;
1,1,1,1,2,2,3,4;
0,0,0,0,0,0,0,0,1;
0,0,0,0,0,0,0,0,1,2;
0,0,0,0,0,0,0,0,0,0,1;
0,0,0,0,0,0,0,0,1,1,2,3;
0,0,0,0,0,0,0,0,0,0,0,0,1;
0,0,0,0,0,0,0,0,0,0,0,0,1,2;
0,0,0,0,0,0,0,0,0,0,0,0,0,0,1;
1,1,1,1,1,1,1,1,2,2,2,2,3,3,4,5;
...

Mirror of A228348. Column 1 is A036987. Also column 1 gives A209229, n >= 1. Right border gives A001511. Positive terms give A228349.

Cf. A001792, A001787, A006519, A011782, A065120, A096996, A129760, A186114, A187816, A187818, A206437, A228350, A228351, A228366, A228367, A228370, A228371, A228525, A228526.

A228348 Triangle of regions and compositions of the positive integers (see Comments lines for definition).

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Aug 21 2013

Triangle read by rows in which row n lists the A006519(n) elements of the row A001511(n) of triangle A065120 followed by A129760(n) zeros, n >= 1.

The equivalent sequence for integer partitions is A193870.

			----------------------------------------------------------
.             Diagram                Triangle
Compositions    of            of compositions (rows)
of 5          regions          and regions (columns)
----------------------------------------------------------
.            _ _ _ _ _
5           |_        |                                 5
1+4         |_|_      |                               1 4
2+3         |_  |     |                             2 0 3
1+1+3       |_|_|_    |                           1 1 0 3
3+2         |_    |   |                         3 0 0 0 2
1+2+2       |_|_  |   |                       1 2 0 0 0 2
2+1+2       |_  | |   |                     2 0 1 0 0 0 2
1+1+1+2     |_|_|_|_  |                   1 1 0 1 0 0 0 2
4+1         |_      | |                 4 0 0 0 0 0 0 0 1
1+3+1       |_|_    | |               1 3 0 0 0 0 0 0 0 1
2+2+1       |_  |   | |             2 0 2 0 0 0 0 0 0 0 1
1+1+2+1     |_|_|_  | |           1 1 0 2 0 0 0 0 0 0 0 1
3+1+1       |_    | | |         3 0 0 0 1 0 0 0 0 0 0 0 1
1+2+1+1     |_|_  | | |       1 2 0 0 0 1 0 0 0 0 0 0 0 1
2+1+1+1     |_  | | | |     2 0 1 0 0 0 1 0 0 0 0 0 0 0 1
1+1+1+1+1   |_|_|_|_|_|   1 1 0 1 0 0 0 1 0 0 0 0 0 0 0 1
.
For the positive integer k consider the first 2^(k-1) rows of triangle, as shown below. The positive terms of the n-th row are the parts of the n-th region of the diagram of regions of the set of compositions of k. The positive terms of the n-th diagonal are the parts of the n-th composition of k, with compositions in colexicographic order.
Triangle begins:
1;
2,1;
1,0,0;
3,2,1,1;
1,0,0,0,0;
2,1,0,0,0,0;
1,0,0,0,0,0,0;
4,3,2,2,1,1,1,1;
1,0,0,0,0,0,0,0,0;
2,1,0,0,0,0,0,0,0,0;
1,0,0,0,0,0,0,0,0,0,0;
3,2,1,1,0,0,0,0,0,0,0,0;
1,0,0,0,0,0,0,0,0,0,0,0,0;
2,1,0,0,0,0,0,0,0,0,0,0,0,0;
1,0,0,0,0,0,0,0,0,0,0,0,0,0,0;
5,4,3,3,2,2,2,2,1,1,1,1,1,1,1,1;
...

Mirror of A228347. Column 1 is A001511. Right border gives A036987. Also right border gives A209229, n >= 1. Positive terms give A228350.

Cf. A001792, A001787, A006519, A011782, A065120, A129760, A187816, A187818, A193870, A206437, A228349, A228351, A228366, A228367, A228370, A228371, A228525, A228526.

A357181 Last run-length of the n-th composition in standard order.

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 1, 3, 1, 1, 2, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 2, 2, 1, 1, 1, 5, 1, 1, 1, 2, 2, 1, 1, 3, 1, 1, 3, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 2, 2, 1, 1, 1, 6, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 2, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1

Offset: 0

Gus Wiseman, Sep 24 2022

nonn

A composition of n is a finite sequence of positive integers summing to n. The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

			Composition 87 in standard order is (2,2,1,1,1), so a(87) = 3.

Gus Wiseman, Statistics, classes, and transformations of standard compositions

See link for sequences related to standard compositions.
For parts instead of run-lengths we have A001511, first A065120.
For Heinz numbers of partitions we have A071178, first A067029.
This is the last part of row n of A333769.
For maximal instead of last we have A357137, minimal A357138.
The first instead of last run-length is A357180.
A051903 gives maximal part of prime signature.
A061395 gives maximal prime index.
A124767 counts runs in standard compositions.
A286470 gives maximal difference of prime indices.
A333766 gives maximal part of standard composition, minimal A333768.
A353847 ranks run-sums of standard compositions.
Cf. A000120, A003754, A029931, A070939, A329395, A356841, A356844, A357134, A357135, A357136.

stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Table[If[n==0,0,Last[Length/@Split[stc[n]]]],{n,0,100}]

A345254 Dispersion of A004754, a rectangular array T(n,k) read by downward antidiagonals.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 9, 10, 7, 16, 17, 18, 11, 12, 32, 33, 34, 19, 20, 13, 64, 65, 66, 35, 36, 21, 14, 128, 129, 130, 67, 68, 37, 22, 15, 256, 257, 258, 131, 132, 69, 38, 23, 24, 512, 513, 514, 259, 260, 133, 70, 39, 40, 25, 1024, 1025, 1026, 515, 516, 261, 134

Offset: 1

J. Parker Shectman, Jun 12 2021

As a sequence, {a(n)} permutes the positive integers. As an array, {T(n,k)} is an interspersion-dispersion or I-D array (refer to Kimberling, 1st linked reference).
The top row of {T(n,k)} is A000079 or powers of two = 1, 2, 4, 8, 16, ....
Except for the leftmost element "1" of the top row, rows of T(n,k) indexed n = 0, 1, 2, ..., consist entirely of even numbers (A005843) for n even and entirely of odd numbers (A005408) for n odd.
The left column (k = 1) of {T(n,k)} comprises a "1" for the top row (n = 0) and A004755(n) = n + 2^(floor(log_2(n)) + 1), for rows n = 1, 2, 3, ....
For rows indexed n = 0, 1, 2, ..., and columns indexed k = 1, 2, 3, ..., T(n,k) is given by T(0,k) = L^(k - 1)(1) and T(n,k) = L^(k - 1) R(n) for n = 1, 2, 3, ..., the image of n under a composition of branching functions L(n) = A004754(n) = n + 2^floor(log_2(n)) and R(n) = A004755(n) = n + 2^(floor(log_2(n)) + 1) (cf. generating tree A059893 and 2nd linked reference).
(Duality with array A054582): Consider A059893 and A000027 as labeled binary trees arranging the positive integers. In latter tree, node labels equal node positions, thus following their natural order. Rows of {T(n,k)} are the labels along maximal straight paths that always branch left in the former tree, while rows of (transposed) array A054582 are the labels along maximal straight paths that always branch left in the latter tree.
Column k of {T(n,k)} comprises the (sorted) labels in the k-th right clade of latter tree, while column k of (transposed) A054582 comprises the (sorted) labels in the k-th right clade of the former tree. This makes the arrays {T(n,k)} and (transposed) A054582 "blade-duals," blade being a contraction of branch-clade ('right clades' explained under tree A345253 and in 2nd link).
Write the positive integers in natural order as a (left-justified) "tetrangle" or "irregular triangle" tableau with 2^t entries on each row t, for t=1, 2, 3, .... Then, columns of the tableau equal rows of {T(n,k)} (2nd link):
   1,
   2,  3,
   4,  5,  6, 7,
   8,  9, 10, 11, 12, 13, 14, 15
  16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31,
  ...
Analogous to A345252, its right-justified tableau of the positive integers in cohorts with lengths the Fibonacci numbers replaced by the above left-justified tableau with powers of two as lengths of the cohorts.
(Mirror duality): A "mirror dual" I-D array or "inverse I-D array" (see Kimberling, 1st linked reference) is obtained by substituting the left-justified tableau by a right-justified tableau and following the identical procedure, or equivalently by mirroring the tree A059893 cited above, i.e., taking maximal straight paths that always branch right in the tree A059893. With two types of duality then, {T(n,k)} forms a quartet of I-D arrays together with its mirror dual, its blade dual (transposed) A054582 and mirror dual A191448 of the latter.
(Para-sequences): Sequences of row and column indices (see Conway-Sloane correspondence under A019586, citing Kimberling). For rows indexed n = 0, 1, 2, ..., and columns indexed k = 1, 2, 3, ..., the row index n of positive integer T(n,k) is A053645(T) and the column index k of positive integer T(n,k) is A065120(T).

			Northwest corner of {T(n,k)}:
       k=1   k=2    k=3     k=4      k=5       k=6
  n=0:   1,    2,     4,      8,      16,       32, ...
  n=1:   3,    5,     9,     17,      33,       65, ...
  n=2:   6,   10,    18,     34,      66,      130, ...
  n=3:   7,   11,    19,     35,      67,      131, ...
  n=4:  12,   20,    36,     68,     132,      260, ...
  ...
Northwest corner of {T(n,k)} in base-2:
        k=1  k=2    k=3     k=4      k=5       k=6
  n=0:  1,   10,    100,    1000,    10000,    100000, ...
  n=1:  11,  101,   1001,   10001,   100001,   1000001, ...
  n=2:  110, 1010,  10010,  100010,  1000010,  10000010, ...
  n=3:  111, 1011,  10011,  100010,  1000011,  10000011, ...
  n=4:  1100,10100, 100100, 1000100, 10000100, 100000100, ...
  ...

Clark Kimberling, Interspersions and dispersions, Proceedings of the American Mathematical Society, 117 (1993) 313-321.
Parker Shectman, A Quilt after Fibonacci, Part 2 of 3: Cohorts, Free Monoids, and Numeration, 2021.
Index entries for sequences that are permutations of the natural numbers

Cf. A000027, A004754, A053645, A005408, A005843, A019586, A054582, A059893, A065120, A139706, A139708, A191448, A345252, A345253.
Rows n=0..1: A000079, A048578.
Columns k=1..2: A004755, A004757.

(*Simplified Formula*)
MatrixForm[Prepend[Table[n + 2^(Floor[Log[2, n]] + k), {n, 1, 4}, {k, 1, 6}], Table[2^(k - 1), {k, 1, 6}]]]
(*Branching Formula*)
MatrixForm[Prepend[Table[NestList[Function[# + 2^(Floor[Log[2, #]])], n + 2^(Floor[Log[2, n]] + 1), 5], {n, 1, 4}], NestList[Function[# + 2^(Floor[Log[2, #]])], 1, 5]]]

T(n, k) = if (n==0, 2^(k-1), n + 2^(log(n)\log(2) + k));
matrix(7, 7, n, k, n--; T(n, k)) \\ Michel Marcus, Jul 30 2021

T(0,k) = 2^(k - 1) and T(n,k) = n + 2^(floor(log_2(n)) + k) for n >= 1.
T(0,k) = L^(k - 1)(1) and T(n,k) = L^(k - 1) R(n) for n = 1, 2, 3, ..., where L(n) = A004754(n) = n + 2^floor(log_2(n)) and R(n) = A004755(n) = n + 2^(floor(log_2(n)) + 1).
Let b(n) = A054582(n-1). Then for all n >= 1, a(n) = A139706(b(n)) and b(n) = A139708(a(n)).

A375379 Value of the power tower formed by the numbers obtained by adding 1 to the terms in the n-th composition (in standard order).

Original entry on oeis.org

1, 2, 3, 4, 4, 9, 8, 16, 5, 16, 27, 81, 16, 512, 256, 65536, 6, 25, 64, 256, 81, 19683, 6561, 43046721, 32, 65536, 134217728, 2417851639229258349412352, 65536

Offset: 0

Pontus von Brömssen, Aug 14 2024

nonn

This is a version of A375378 that does not allow 1's in the power towers.
It is natural to define a(0) = 1.
The number k >= 1 appears A294336(k) times as a term in this sequence.

			For n = 31, the 31st composition is (1, 1, 1, 1, 1), so a(31) = 2^2^2^2^2 = 2^65536.
For n = 37, the 37th composition is (3,2,1), so a(37) = 4^3^2 = 4^9 = 262144.

Pontus von Brömssen, Table of n, a(n) for n = 0..30

Cf. A053645, A065120, A066099 (compositions in standard order), A294336, A375378.

a(n) = (A065120(n)+1)^a(A053645(n)) for n >= 1.

cp's OEIS Frontend

A123639 Consider the 2^n compositions of n and count only those ending in an even part.

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A228349 Triangle read by rows: T(j,k) is the k-th part in nondecreasing order of the j-th region of the set of compositions (ordered partitions) of n in colexicographic order, if 1<=j<=2^(n-1) and 1<=k<=A006519(j).

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A357180 First run-length of the n-th composition in standard order.

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A375139 Numbers k such that the leaders of strictly increasing runs in the k-th composition in standard order are not weakly decreasing.

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A123638 Consider the 2^n compositions of n and count only those ending in an odd part with row sum A001045.

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A228347 Triangle of regions and compositions of the positive integers (see Comments lines for definition).

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A228348 Triangle of regions and compositions of the positive integers (see Comments lines for definition).

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A357181 Last run-length of the n-th composition in standard order.

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A345254 Dispersion of A004754, a rectangular array T(n,k) read by downward antidiagonals.

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