A164894 -id:A164894 - cp's OEIS frontend

A356956 Numbers k such that the k-th composition in standard order is a gapless interval (in increasing order).

0, 1, 2, 4, 6, 8, 16, 20, 32, 52, 64, 72, 128, 256, 272, 328, 512, 840, 1024, 1056, 2048, 2320, 4096, 4160, 8192, 10512, 16384, 16512, 17440, 26896, 32768, 65536, 65792, 131072, 135232, 148512, 262144, 262656, 524288, 672800, 1048576, 1049600, 1065088, 1721376

Offset: 1

Gus Wiseman, Sep 24 2022

nonn

An interval such as {3,4,5} is a set of positive integers with all differences of adjacent elements equal to 1.

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 and corresponding intervals begin:
        0: ()
        1: (1)
        2: (2)
        4: (3)
        6: (1,2)
        8: (4)
       16: (5)
       20: (2,3)
       32: (6)
       52: (1,2,3)
       64: (7)
       72: (3,4)
      128: (8)
      256: (9)
      272: (4,5)
      328: (2,3,4)
      512: (10)
      840: (1,2,3,4)

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

See link for sequences related to standard compositions.
These compositions are counted by A001227.
An unordered version is A073485, non-strict A073491 (complement A073492).
The initial version is A164894, non-strict A356843 (unordered A356845).
The non-strict version is A356841, initial A333217, counted by A107428.
A066311 lists gapless numbers.
A356230 ranks gapless factorization lengths, firsts A356603.
A356233 counts factorizations into gapless numbers.
A356844 ranks compositions with at least one 1.
Cf. A053251, A055932, A073493, A132747, A137921, A286470, A356224, A356842.

stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
chQ[y_]:=Length[y]<=1||Union[Differences[y]]=={1};
Select[Range[0,1000],chQ[stc[#]]&]

A382288 Number of records in the n-th composition in standard order.

Original entry on oeis.org

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

Offset: 0

John Tyler Rascoe, Mar 20 2025

Here a record is a part of the composition that is greater than all parts before it, reading left to right. The first part of any nonempty composition is a record so a(n) >= 1 for n > 0. See A066099 for the standard order of integer compositions.

The first appearance of k occurs at n = A164894(k) for k > 0.

			The 883rd composition is (1, 2, 1, 1, 3, 1, 1) with records 1, 2, and 3; so a(883) = 3.
                          ^  ^        ^

Cf. A066099, A124762, A124767, A124768, A164894, A333381, A333382, A382312.

def comp(n):
    return # see A357625
def A382288(n):
    r,c = 0,0
    for i in comp(n):
        if i > r:
            c += 1
            r = i
    return c

a(A164894(n)) = n for n > 0.

A385373 Number of solid partitions with multiplicities (1, ..., n).

Original entry on oeis.org

1, 1, 6, 138, 14049, 6851919

Offset: 0

John Tyler Rascoe, Jun 27 2025

A solid partition with d distinct parts (p_1^(k_1) > p_2^(k_2) > ... > p_d^(k_d)) has the multiset of multiplicities (k_1, k_2, ..., k_d).

Alternatively, a(n) is the number of chains of plane partitions ordered by inclusion, comprised of n consecutive triangular numbers starting with 1.

			For n = 2 a solid partition having multiplicities (1,2) has two distinct parts (a,b^2) with a < b, and there are 6 ways to arrange these parts.

John Tyler Rascoe, Python program.

Cf. A000217, A000219, A000293, A164894, A379277.

Python
```
# see Links
```

a(n) = A379277(A164894(n)) for n > 0.

A386552 Concatenate powers of 10.

Original entry on oeis.org

1, 110, 110100, 1101001000, 110100100010000, 110100100010000100000, 1101001000100001000001000000, 110100100010000100000100000010000000, 110100100010000100000100000010000000100000000, 1101001000100001000001000000100000001000000001000000000

Offset: 0

Jason Bard, Jul 25 2025

Binary version of A045507. Base-2 representation of A164894.

Concatenate first A000217(n+1) terms of A010054.

Jason Bard, Table of n, a(n) for n = 0..43

Cf. A000096, A000217, A010054, A011557, A045507, A051639, A101305, A133082, A164894, A242646.

a:= proc(n) option remember;
      `if`(n<0, 0, parse(cat(a(n-1), 10^n)))
    end:
seq(a(n), n=0..10);  # Alois P. Heinz, Jul 28 2025

a[0] = 1; a[n_] := a[n - 1]*10^(n+1) + 10^n; Table[a[n], {n, 0, 9}]

def A386552(n): return 10**n*sum(10**(k*((n<<1)-k+1)>>1) for k in range(n+1)) # Chai Wah Wu, Aug 05 2025

a(n) = Sum_{k=1..n+1} 10^A133082(k,n+2).
a(n) = A101305(n) + 10^A000096(n).
For n >= 1, a(n) = 10^(n+1)*a(n-1)+10^n.
Number of digits in a(n) is A000217(n+1).

cp's OEIS Frontend

A356956 Numbers k such that the k-th composition in standard order is a gapless interval (in increasing order).

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A382288 Number of records in the n-th composition in standard order.

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A385373 Number of solid partitions with multiplicities (1, ..., n).

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Python

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A386552 Concatenate powers of 10.

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Maple

Mathematica

Python

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