A228351 Triangle read by rows in which row n lists the compositions (ordered partitions) of n (see Comments lines for definition).

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

Offset: 1

Author

Omar E. Pol, Aug 30 2013

Keywords

• nonn
• tabf

Comments

The representation of the compositions (for fixed n) is as lists of parts, the order between individual compositions (for the same n) is (list-)reversed co-lexicographic. - Joerg Arndt, Sep 02 2013

Dropping the "(list-)reversed" in the comment above gives A228525.

The equivalent sequence for partitions is A026792.

This sequence lists (without repetitions) all finite compositions, in such a way that, if [P_1, ..., P_r] denotes the composition occupying the n-th position in the list, then (((2*n/2^(P_1)-1)/2^(P_2)-1)/...)/2^(P_r)-1 = 0. - Lorenzo Sauras Altuzarra, Jan 22 2020

The k-th composition in the list is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, and taking first differences. Reversing again gives A066099, which is described as the standard ordering. Both sequences define a bijective correspondence between nonnegative integers and integer compositions. - Gus Wiseman, Apr 01 2020

It follows from the previous comment that A000120(k) is the length of the k-th composition that is listed by this sequence (recall that A000120(k) is the number of 1's in the binary expansion of k). - Lorenzo Sauras Altuzarra, Sep 29 2020

Examples

			Illustration of initial terms:
-----------------------------------
n  j     Diagram     Composition j
-----------------------------------
.         _
1  1     |_|         1;
.         _ _
2  1     |_  |       2,
2  2     |_|_|       1, 1;
.         _ _ _
3  1     |_    |     3,
3  2     |_|_  |     1, 2,
3  3     |_  | |     2, 1,
3  4     |_|_|_|     1, 1, 1;
.         _ _ _ _
4  1     |_      |   4,
4  2     |_|_    |   1, 3,
4  3     |_  |   |   2, 2,
4  4     |_|_|_  |   1, 1, 2,
4  5     |_    | |   3, 1,
4  6     |_|_  | |   1, 2, 1,
4  7     |_  | | |   2, 1, 1,
4  8     |_|_|_|_|   1, 1, 1, 1;
.
Triangle begins:
[1];
[2],[1,1];
[3],[1,2],[2,1],[1,1,1];
[4],[1,3],[2,2],[1,1,2],[3,1],[1,2,1],[2,1,1],[1,1,1,1];
[5],[1,4],[2,3],[1,1,3],[3,2],[1,2,2],[2,1,2],[1,1,1,2],[4,1],[1,3,1],[2,2,1],[1,1,2,1],[3,1,1],[1,2,1,1],[2,1,1,1],[1,1,1,1,1];
...
For example [1,2] occupies the 5th position in the corresponding list of compositions and indeed (2*5/2^1-1)/2^2-1 = 0. - _Lorenzo Sauras Altuzarra_, Jan 22 2020
12 --binary expansion--> [1,1,0,0] --reverse--> [0,0,1,1] --positions of 1's--> [3,4] --prepend 0--> [0,3,4] --first differences--> [3,1]. - _Lorenzo Sauras Altuzarra_, Sep 29 2020
		

Links

• Peter Kagey, Table of n, a(n) for n = 1..10000
• Mikhail Kurkov, Comments on A228351
• Index entries for sequences that are related to compositions

Crossrefs

Row n has length A001792(n-1). Row sums give A001787, n >= 1.

Cf. A000120 (binary weight), A001511, A006519, A011782, A026792, A065120.

A related ranking of finite sets is A048793/A272020.

All of the following consider the k-th row to be the k-th composition, ignoring the coarser grouping by sum.

- Indices of weakly increasing rows are A114994.

- Indices of weakly decreasing rows are A225620.

- Indices of strictly decreasing rows are A333255.

- Indices of strictly increasing rows are A333256.

- Indices of reversed interval rows A164894.

- Indices of interval rows are A246534.

- Indices of strict rows are A233564.

- Indices of constant rows are A272919.

- Indices of anti-run rows are A333489.

- Row k has A124767(k) runs and A333381(k) anti-runs.

- Row k has GCD A326674(k) and LCM A333226(k).

- Row k has Heinz number A333219(k).

Cf. A000120, A029931, A035327, A070939, A233249, A333217, A333218, A333220, A333227, A333627, A333628.

Equals A163510+1, termwise.

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. A337260 (decreasing length, then colexicographic).

Cf. A108244 (decreasing length, then reverse colexicographic).

Cf. A228369 (lexicographic).

Cf. A066099 (reverse lexicographic).

Cf. A228525 (colexicographic).

Programs

Haskell

a228351 n = a228351_list !! (n - 1)
a228351_list = concatMap a228351_row [1..]
a228351_row 0 = []
a228351_row n = a001511 n : a228351_row (n `div` 2^(a001511 n))
-- Peter Kagey, Jun 27 2016

Maple

# Program computing the sequence:
A228351 := proc(n) local c, k, L, N: L, N := [], [seq(2*r, r = 1 .. n)]: for k in N do c := 0: while k != 0 do if gcd(k, 2) = 2 then k := k/2: c := c+1: else L := [op(L), op(c)]: k := k-1: c := 0: fi: od: od: L[n]: end: # Lorenzo Sauras Altuzarra, Jan 22 2020
# Program computing the list of compositions:
List := proc(n) local c, k, L, M, N: L, M, N := [], [], [seq(2*r, r = 1 .. 2^n-1)]: for k in N do c := 0: while k != 0 do if gcd(k, 2) = 2 then k := k/2: c := c+1: else L := [op(L), c]: k := k-1: c := 0: fi: od: M := [op(M), L]: L := []: od: M: end: # Lorenzo Sauras Altuzarra, Jan 22 2020

Mathematica

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Table[Differences[Prepend[bpe[n],0]],{n,0,30}] (* Gus Wiseman, Apr 01 2020 *)

Python

from itertools import count, islice
def A228351_gen(): # generator of terms
    for n in count(1):
        k = n
        while k:
            yield (s:=(~k&k-1).bit_length()+1)
            k >>= s
A228351_list = list(islice(A228351_gen(),30)) # Chai Wah Wu, Jul 17 2023