A242628 -id:A242628 - cp's OEIS frontend

A253566 Permutation of natural numbers: a(n) = A243071(A122111(n)).

0, 1, 2, 3, 4, 6, 8, 7, 5, 12, 16, 14, 32, 24, 10, 15, 64, 13, 128, 28, 20, 48, 256, 30, 9, 96, 11, 56, 512, 26, 1024, 31, 40, 192, 18, 29, 2048, 384, 80, 60, 4096, 52, 8192, 112, 22, 768, 16384, 62, 17, 25, 160, 224, 32768, 27, 36, 120, 320, 1536, 65536, 58, 131072, 3072, 44, 63, 72, 104, 262144, 448, 640, 50, 524288, 61, 1048576, 6144, 21

Offset: 1

Antti Karttunen, Jan 03 2015

nonn

Note the indexing: domain starts from one, while the range includes also zero. See also comments in A253564.

The a(n)-th composition in standard order (graded reverse-lexicographic, A066099) is one plus the first differences of the weakly increasing sequence of prime indices of n with 1 prepended. See formula for a simplification. The triangular form is A358169. The inverse is A253565. Not prepending 1 gives A358171. For Heinz numbers instead of standard compositions we have A325351 (without prepending A325352). - Gus Wiseman, Dec 23 2022

			From _Gus Wiseman_, Dec 23 2022: (Start)
This represents the following bijection between partitions and compositions. The reversed prime indices of n together with the a(n)-th composition in standard order are:
   1:        () -> ()
   2:       (1) -> (1)
   3:       (2) -> (2)
   4:     (1,1) -> (1,1)
   5:       (3) -> (3)
   6:     (2,1) -> (1,2)
   7:       (4) -> (4)
   8:   (1,1,1) -> (1,1,1)
   9:     (2,2) -> (2,1)
  10:     (3,1) -> (1,3)
  11:       (5) -> (5)
  12:   (2,1,1) -> (1,1,2)
  13:       (6) -> (6)
  14:     (4,1) -> (1,4)
  15:     (3,2) -> (2,2)
  16: (1,1,1,1) -> (1,1,1,1)
(End)

Antti Karttunen, Table of n, a(n) for n = 1..1024
Index entries for sequences that are permutations of the natural numbers

Inverse: A253565.
Cf. A122111, A243071, A253564, A054429, A336120, A336125.
Applying A000120 gives A001222.
A reverse version is A156552, inverse essentially A005940.
The inverse is A253565, triangular form A242628.
The triangular form is A358169.
A048793 gives partial sums of reversed standard comps, Heinz number A019565.
A066099 lists standard compositions, lengths A000120, sums A070939.
A112798 list prime indices, sum A056239.
A358134 gives partial sums of standard compositions, Heinz number A358170.
Cf. A029837, A241916, A243055, A287352, A325390, A355534, A358171, A359042.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
stcinv[q_]:=Total[2^(Accumulate[Reverse[q]])]/2;
stcinv/@Table[Differences[Prepend[primeMS[n],1]]+1,{n,100}] (* Gus Wiseman, Dec 23 2022 *)

(define (A253566 n) (A243071 (A122111 n)))

a(n) = A243071(A122111(n)).
As a composition of other permutations:
a(n) = A054429(A253564(n)).
a(n) = A336120(n) + A336125(n). - Antti Karttunen, Jul 18 2020
If 2n = Product_{i=1..k} prime(x_i) then a(n) = Sum_{i=1..k-1} 2^(x_k-x_{k-i}+i-1). - Gus Wiseman, Dec 23 2022

A241918 Table of partitions where the ordering is based on the modified partial sums of the exponents of primes in the prime factorization of n.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, May 03 2014, based on Marc LeBrun's Jan 11 2006 message on SeqFan mailing list

a(1) = 0 by convention (stands for an empty partition).

For n >= 2, A203623(n-1)+2 gives the index to the beginning of row n and for n>=1, A203623(n)+1 is the index to the end of row n.

			Table begins:
Row     Partition
[ 1]    0;         (stands for empty partition)
[ 2]    1;         (as 2 = 2^1)
[ 3]    1,1;       (as 3 = 2^0 * 3^1)
[ 4]    2;         (as 4 = 2^2)
[ 5]    1,1,1;     (as 5 = 2^0 * 3^0 * 5^1)
[ 6]    2,2;       (as 6 = 2^1 * 3^1)
[ 7]    1,1,1,1;   (as 7 = 2^0 * 3^0 * 5^0 * 7^1)
[ 8]    3;         (as 8 = 2^3)
[ 9]    1,2;       (as 9 = 2^0 * 3^2)
[10]    2,2,2;     (as 10 = 2^1 * 3^0 * 5^1)
[11]    1,1,1,1,1;
[12]    3,3;
[13]    1,1,1,1,1,1;
[14]    2,2,2,2;
[15]    1,2,2;     (as 15 = 2^0 * 3^1 * 5^1)
[16]    4;
[17]    1,1,1,1,1,1,1;
[18]    2,3;       (as 18 = 2^1 * 3^2)
etc.
If n is 2^k (k>=1), then the partition is a singleton {k}, otherwise, add one to the exponent of 2 (= A007814(n)), and subtract one from the exponent of the greatest prime dividing n (= A071178(n)), leaving the intermediate exponents as they are, and then take partial sums of all, thus resulting for e.g. 15 = 2^0 * 3^1 * 5^1 the modified sequence of exponents {0+1, 1, 1-1} -> {1,1,0}, whose partial sums {1,1+1,1+1+0} -> {1,2,2} give the corresponding partition at row 15.

Antti Karttunen, Table of n, a(n) for n = 1..10081; rows 1..521 flattened.
Marc LeBrun's original "crazy order" mapping for partitions (Copy of Marc's Jan 11 2006 message in OEIS Wiki)

For n>=2, the length of row n is given by A061395(n).
Cf. also A067255, A203623, A241914.
Other tables of partitions: A112798 (also based on prime factorization), A227739, A242628 (encoded in the binary representation of n), and A036036-A036037, A080576-A080577, A193073 for various lexicographical orderings.
Permutation A241909 maps between order of partitions employed here, and the order employed in A112798.
Permutation A122111 is induced when partitions in this list are conjugated.
A241912 gives the row numbers for which the corresponding rows in A112798 and here are the conjugate partitions of each other.

Table[If[n == 1, {0}, Function[s, Function[t, Accumulate[If[Length@ t < 2, {0}, Join[{1}, ConstantArray[0, Length@ t - 2], {-1}]] + ReplacePart[t, Map[#1 -> #2 & @@ # &, s]]]]@ ConstantArray[0, Transpose[s][[1, -1]]]][FactorInteger[n] /. {p_, e_} /; p > 0 :> {PrimePi@ p, e}]], {n, 31}] // Flatten (* Michael De Vlieger, May 12 2017 *)

If A241914(n)=0 and A241914(n+1)=0, a(n) = A067255(n); otherwise, if A241914(n)=0 and A241914(n+1)>0, a(n) = A067255(n)+1; otherwise, if A241914(n)>0 and A241914(n+1)=0, a(n) = a(n-1) + A067255(n) - 1, otherwise, when A241914(n)>0 and A241914(n+1)>0, a(n) = a(n-1) + A067255(n).

A358135 Difference of first and last parts of the n-th composition in standard order.

Original entry on oeis.org

0, 0, 0, 0, -1, 1, 0, 0, -2, 0, -1, 2, 0, 1, 0, 0, -3, -1, -2, 1, -1, 0, -1, 3, 0, 1, 0, 2, 0, 1, 0, 0, -4, -2, -3, 0, -2, -1, -2, 2, -1, 0, -1, 1, -1, 0, -1, 4, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 0, -5, -3, -4, -1, -3, -2, -3, 1, -2, -1, -2, 0, -2

Offset: 1

Gus Wiseman, Oct 31 2022

sign

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.

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

See link for sequences related to standard compositions.
The first and last parts are A065120 and A001511.
This is the first minus last part of row n of A066099.
The version for Heinz numbers of partitions is A243055.
Row sums of A358133.
The partial sums of standard compositions are A358134, adjusted A242628.
A011782 counts compositions.
A333766 and A333768 give max and min in standard compositions, diff A358138.
A351014 counts distinct runs in standard compositions.
Cf. A000120, A001511, A029837, A029931, A048896, A058891, A070939, A133494, A329395, A357187.

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

a(n) = A001511(n) - A065120(n).

A358133 Triangle read by rows whose n-th row lists the first differences of the n-th composition in standard order (row n of A066099).

Original entry on oeis.org

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

Offset: 3

Gus Wiseman, Oct 31 2022

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.

			Triangle begins (dots indicate empty rows):
   1:   .
   2:   .
   3:   0
   4:   .
   5:  -1
   6:   1
   7:   0  0
   8:   .
   9:  -2
  10:   0
  11:  -1  0
  12:   2
  13:   1 -1
  14:   0  1
  15:   0  0  0

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

See link for sequences related to standard compositions.
First differences of rows of A066099.
The version for Heinz numbers of partitions is A355536, ranked by A253566.
The partial sums instead of first differences are A358134.
Row sums are A358135.
A011782 counts compositions.
A351014 counts distinct runs in standard compositions.
Cf. A000120, A001511, A029837, A029931, A048896, A070939, A133494, A242628, A357135, A357187.

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

A358170 Heinz number of the partial sums of the n-th composition in standard order (A066099).

Original entry on oeis.org

1, 2, 3, 6, 5, 15, 10, 30, 7, 35, 21, 105, 14, 70, 42, 210, 11, 77, 55, 385, 33, 231, 165, 1155, 22, 154, 110, 770, 66, 462, 330, 2310, 13, 143, 91, 1001, 65, 715, 455, 5005, 39, 429, 273, 3003, 195, 2145, 1365, 15015, 26, 286, 182, 2002, 130, 1430, 910, 10010

Offset: 0

Gus Wiseman, Dec 20 2022

nonn

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 Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

			The terms together with their prime indices begin:
           1: {}
           2: {1}
           3: {2}
           6: {1,2}
           5: {3}
          15: {2,3}
          10: {1,3}
          30: {1,2,3}
           7: {4}
          35: {3,4}
          21: {2,4}
         105: {2,3,4}
          14: {1,4}
          70: {1,3,4}
          42: {1,2,4}
         210: {1,2,3,4}

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

See link for sequences related to standard compositions.
Applying A001221 or A001222 gives A000120.
The image is A005117 (squarefree numbers).
The reverse version is A019565, triangular version A048793.
Greatest prime index of a(n) is A029837 or A070939.
Least prime index of a(n) is A065120.
The adjusted version is A253565, inverse A253566, reverse A005940.
These are the Heinz numbers of the rows of A358134.
Sum of prime indices of a(n) is A359042.
A066099 lists standard compositions.
A112798 list prime indices, sum A056239.
Cf. A000720, A001511, A029931, A059893, A061395, A241916, A242628, A355536, A358133, A358135, A358137.

stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Times@@Prime/@#&/@Table[Accumulate[stc[n]],{n,0,100}]

A112531 Triangle read by rows which lists compositions having at least one part equal to 1.

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 3, 1, 2, 3, 6, 4, 3, 6, 4, 1, 2, 3, 6, 4, 6, 12, 12, 5, 3, 6, 12, 10, 4, 10, 5, 1, 2, 3, 6, 4, 6, 12, 12, 5, 6, 12, 24, 20, 12, 30, 20, 6, 3, 6, 12, 10, 12, 30, 30, 15, 4, 10, 20, 20, 5, 15, 6, 1

Offset: 1

Alford Arnold, Sep 10 2005

Consider partitions listed in the order given by A241596 and A242628. Omit any partition not containing 1 as a part. Write down the number of compositions (= ordered partitions) corresponding to this partition.
Row sums give A112532; which are the first differences of A047970.
Row lengths give A011782.

			The partitions (see A241596) begin 1 2 11 3 22 21 111 4 33 32 222 31 221 211 1111 ...
After omitting partitions with no part equal to 1, we have
1 11 21 111 31 221 211 1111 ...
which give rise to 1 1 2 1 2 3 3 1 ... compositions.
The resulting triangle of compositions begins:
1;
1;
2, 1;
2, 3, 3, 1;
2, 3, 6, 4, 3, 6, 4, 1;
2, 3, 6, 4, 6, 12, 12, 5, 3, 6, 12, 10, 4, 10, 5, 1;
2, 3, 6, 4, 6, 12, 12, 5, 6, 12, 24, 20, 12, 30, 20, 6, 3, 6, 12, 10, 12, 30, 30, 15, 4, 10, 20, 20, 5, 15, 6, 1;
...

Cf. A000079, A011782, A047970, A089246, A112532, A241596, A242628.

Edited by N. J. A. Sloane, May 19 2014 based on postings to the Sequence Fans Mailing List by Peter Luschny, Jonas Wallgren, Arie Groeneveld, and Franklin T. Adams-Watters.

A241596 Partitions listed by alternately incrementing each part and appending a 1.

Original entry on oeis.org

1, 2, 11, 3, 22, 21, 111, 4, 33, 32, 222, 31, 221, 211, 1111, 5, 44, 43, 333, 42, 332, 322, 2222, 41, 331, 321, 2221, 311, 2211, 2111, 11111, 6, 55, 54, 444, 53, 443, 433, 3333, 52, 442, 432, 3332, 422, 3322, 3222, 22222, 51, 441, 431, 3331, 421, 3321, 3221, 22221, 411, 3311, 3211, 22211, 3111, 22111, 21111, 111111

Offset: 1

N. J. A. Sloane, May 19 2014

Start with S_0 = {1}.
Thereafter, S_{n+1} consists of the partitions in S_n with all parts incremented by 1, together with all partitions in S_n with an additional part of 1.
From Franklin T. Adams-Watters, May 19 2014:
a(n) can be defined in terms of the binary expansion of n. Start with the partition [1]. Now process the bits of n from right to left, excluding the leading 1. For a zero bit, increase each number in the partition by 1; for a one bit, add a part of size 1. For example, for n=11, binary 1011, we get 1 -> 11 -> 111 -> 222 = a(11).
Row n consists of all partitions with hook size (maximum part + number of parts - 1) equal to n.
This sequence will eventually fail because digits greater than 9 are needed.

			The partitions appear in the following order:
S_0 = 1,
S_1 = 2, 11,
S_2 = 3, 22, 21, 111,
S_3 = 4, 33, 32, 222, 31, 221, 211, 1111,
S_4 = 5, 44, 43, 333, 42, 332, 322, 2222, 41, 331, 321, 2221, 311, 2211, 2111, 11111,
...

Arie Groeneveld, Posting to Sequence Fans List, May 19 2014

Alois P. Heinz, Rows n = 1..9, flattened

See A242628 for another version of this list of partitions.

Cf. A125106, A240837, A112531, A241597 (compositions).

b:= proc(n) option remember; `if`(n=1, [[1]],
      [map(x-> map(y-> y+1, x), b(n-1))[],
       map(x-> [x[], 1], b(n-1))[]])
    end:
T:= n-> map(x-> parse(cat(x[])), b(n))[]:
seq(T(n), n=1..6);

Typos corrected by Alois P. Heinz, Sep 25 2015

A358138 Difference between maximum and minimum part in the n-th composition in standard order.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Oct 31 2022

nonn

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.

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

See link for sequences related to standard compositions.
The first and last parts are A065120 and A001511, difference A358135.
This is the maximum minus minimum part in row n of A066099.
The version for Heinz numbers of partitions is A243055.
The maximum and minimum parts are A333766 and A333768.
The partial sums of standard compositions are A358134, adjusted A242628.
A011782 counts compositions.
A351014 counts distinct runs in standard compositions.
Cf. A000120, A029837, A029931, A048896, A058891, A070939, A133494, A329395, A357187, A358133.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Table[Max[stc[n]]-Min[stc[n]],{n,1,100}]

a(n) = A333766(n) - A333768(n).

A358171 The a(n)-th composition in standard order (A066099) is the first differences plus one of the prime indices of n (A112798).

Original entry on oeis.org

0, 0, 0, 1, 0, 2, 0, 3, 1, 4, 0, 6, 0, 8, 2, 7, 0, 5, 0, 12, 4, 16, 0, 14, 1, 32, 3, 24, 0, 10, 0, 15, 8, 64, 2, 13, 0, 128, 16, 28, 0, 20, 0, 48, 6, 256, 0, 30, 1, 9, 32, 96, 0, 11, 4, 56, 64, 512, 0, 26, 0, 1024, 12, 31, 8, 40, 0, 192, 128, 18, 0, 29, 0

Offset: 1

Gus Wiseman, Dec 21 2022

nonn

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.

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The prime indices of 36 are {1,1,2,2}, with first differences plus one (1,2,1), which is the 13th composition in standard order, so a(36) = 13.

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

See link for sequences related to standard compositions.
Prepend 1 to indices: A253566 (cf. A358169), inverse A253565 (cf. A242628).
Taking Heinz number instead of standard composition number gives A325352.
These compositions minus one are listed by A355536, sums A243055.
A001222 counts prime indices, distinct A001221.
A066099 lists standard compositions, lengths A000120, sums A070939.
A112798 lists prime indices, sum A056239.
A355534 = augmented diffs. of rev. prime indices, Heinz numbers A325351.
Cf. A000720, A005940, A055932, A057335, A066312, A286470, A287352, A325390.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
stcinv[q_]:=Total[2^(Accumulate[Reverse[q]])]/2;
Table[stcinv[Differences[primeMS[n]]+1],{n,100}]

cp's OEIS Frontend

A253566 Permutation of natural numbers: a(n) = A243071(A122111(n)).

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A241918 Table of partitions where the ordering is based on the modified partial sums of the exponents of primes in the prime factorization of n.

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A358135 Difference of first and last parts of the n-th composition in standard order.

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A358133 Triangle read by rows whose n-th row lists the first differences of the n-th composition in standard order (row n of A066099).

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A358170 Heinz number of the partial sums of the n-th composition in standard order (A066099).

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A112531 Triangle read by rows which lists compositions having at least one part equal to 1.

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A241596 Partitions listed by alternately incrementing each part and appending a 1.

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A358138 Difference between maximum and minimum part in the n-th composition in standard order.

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