A335449 -id:A335449 - cp's OEIS frontend

A374768 Numbers k such that the leaders of weakly increasing runs in the k-th composition in standard order (A066099) are distinct.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 26, 28, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 46, 47, 48, 50, 52, 56, 58, 60, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 78, 79, 80, 81

Offset: 1

Gus Wiseman, Jul 19 2024

nonn

First differs from A335467 in having 166, corresponding to the composition (2,3,1,2).
The leaders of weakly increasing runs in a sequence are obtained by splitting it into maximal weakly 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 4444th composition in standard order is (4,2,2,1,1,3), with weakly increasing runs ((4),(2,2),(1,1,3)), with leaders (4,2,1), so 4444 is in the sequence.

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

These are the positions of strict rows in A374629 (which has sums A374630).
Compositions of this type are counted by A374632, increasing A374634.
Identical instead of distinct leaders are A374633, counted by A374631.
For leaders of anti-runs we have A374638, counted by A374518.
For leaders of strictly increasing runs we have A374698, counted by A374687.
For leaders of weakly decreasing runs we have A374701, counted by A374743.
For leaders of strictly decreasing runs we have A374767, counted by A374761.
A011782 counts compositions.
A238130, A238279, A333755 count compositions by number of runs.
All of the following pertain to compositions in standard order:
- Ones are counted by A000120.
- Sum is A029837 (or sometimes A070939).
- Parts are listed by A066099.
- Length is A070939.
- Adjacent equal pairs are counted by A124762, unequal A333382.
- Number of max runs: A124765, A124766, A124767, A124768, A124769, A333381.
- Ranks of strict compositions are A233564.
- Ranks of constant compositions are A272919.
- Ranks of anti-run compositions are A333489, counted by A003242.
- Run-length transform is A333627.
- Run-compression transform is A373948, sum A373953, excess A373954.
Cf. A065120, A188920, A189076, A238343, A333213, A335449, A373949, A274174, A374249, A374635, A374637.

stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,166],UnsameQ@@First/@Split[stc[#],LessEqual]&]

A333175 If n = Product (p_j^k_j) then a(n) = Sum (a(n/p_j^k_j)), with a(1) = 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 6, 1, 1, 2, 2, 2, 2, 1, 2, 2, 2, 1, 6, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 6, 1, 2, 2, 1, 2, 6, 1, 2, 2, 6, 1, 2, 1, 2, 2, 2, 2, 6, 1, 2, 1, 2, 1, 6, 2, 2, 2, 2, 1, 6, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2

Offset: 1

Ilya Gutkovskiy, Mar 11 2020

nonn

Number of ordered prime factorizations of radical of n.

Number of permutations of the prime indices of n (counting multiplicity) avoiding the patterns (1,2,1) and (2,1,2). These are permutations with all equal parts contiguous. Depends only on sorted prime signature (A118914). - Gus Wiseman, Jun 27 2020

			From _Gus Wiseman_, Jun 27 2020 (Start)
The a(n) permutations of prime indices for n = 2, 12, 60:
  (1)  (112)  (1123)
       (211)  (1132)
              (2113)
              (2311)
              (3112)
              (3211)
(End)

Robert Israel, Table of n, a(n) for n = 1..10000
Wikipedia, Permutation pattern
Gus Wiseman, Sequences counting and ranking compositions by the patterns they match or avoid.

Cf. A000142, A000961 (positions of 1's), A001221, A050363, A066504, A069513, A064372, A093320, A292586.
Dominates A335451.
Permutations of prime indices are A008480.
Unsorted prime signature is A124010. Sorted prime signature is A118914.
(1,2,1)-avoiding permutations of prime indices are A335449.
(2,1,2)-avoiding permutations of prime indices are A335450.
(1,2,1) or (2,1,2)-matching permutations of prime indices are A335460.
(1,2,1) and (2,1,2)-matching permutations of prime indices are A335462.
Cf. A056239, A112798, A181796, A333221, A335452, A335463, A335521.

f:= n -> nops(numtheory:-factorset(n))!:
map(f, [$1..100]); # Robert Israel, Mar 12 2020

a[1] = 1; a[n_] := a[n] = Plus @@ (a[n/#[[1]]^#[[2]]] & /@ FactorInteger[n]); Table[a[n], {n, 1, 100}]
a[1] = 1; a[n_] := a[n] = Sum[If[GCD[n/d, d] == 1 && d < n, Boole[PrimePowerQ[n/d]] a[d], 0], {d, Divisors[n]}]; Table[a[n], {n, 1, 100}]
Table[PrimeNu[n]!, {n, 1, 100}]

a(1) = 1; a(n) = Sum_{d|n, d < n, gcd(d, n/d) = 1} A069513(n/d) * a(d).

a(n) = A000142(A001221(n)).

A335446 Number of (1,2,1)-matching permutations of the prime indices of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 2, 0, 0, 0, 1, 1, 0, 0, 3, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 6, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 7, 0, 0, 0, 1, 0, 0, 0, 3, 0, 0, 0, 6, 0, 0, 0

Offset: 1

Gus Wiseman, Jun 13 2020

nonn

Depends only on unsorted prime signature (A124010), but not only on sorted prime signature (A118914).
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.
We define a pattern to be a finite sequence covering an initial interval of positive integers. Patterns are counted by A000670. A sequence S is said to match a pattern P if there is a not necessarily contiguous subsequence of S whose parts have the same relative order as P. For example, (3,1,1,3) matches (1,1,2), (2,1,1), and (2,1,2), but avoids (1,2,1), (1,2,2), and (2,2,1).

			The a(n) permutations for n = 12, 24, 36, 60, 72, 90, 120, 144:
  (121)  (1121)  (1212)  (1213)  (11212)  (1232)  (11213)  (111212)
         (1211)  (1221)  (1231)  (11221)  (2132)  (11231)  (111221)
                 (2121)  (1312)  (12112)  (2312)  (11312)  (112112)
                         (1321)  (12121)  (2321)  (11321)  (112121)
                         (2131)  (12211)          (12113)  (112211)
                         (3121)  (21121)          (12131)  (121112)
                                 (21211)          (12311)  (121121)
                                                  (13112)  (121211)
                                                  (13121)  (122111)
                                                  (13211)  (211121)
                                                  (21131)  (211211)
                                                  (21311)  (212111)
                                                  (31121)
                                                  (31211)

Wikipedia, Permutation pattern
Gus Wiseman, Sequences counting and ranking compositions by the patterns they match or avoid.

Positions of zeros are A065200.
The avoiding version is A335449.
Patterns are counted by A000670.
Permutations of prime indices are counted by A008480.
Unimodal permutations of prime indices are counted by A332288.
(1,2,1) and (2,1,2)-avoiding permutations of prime indices are A333175.
STC-numbers of permutations of prime indices are A333221.
Patterns matched by standard compositions are counted by A335454.
(1,2,1) or (2,1,2)-matching permutations of prime indices are A335460.
(1,2,1) and (2,1,2)-matching permutations of prime indices are A335462.
Dimensions of downsets of standard compositions are A335465.
(1,2,1)-matching compositions are ranked by A335466.
(1,2,1)-matching compositions are counted by A335470.
(1,2,1)-matching patterns are counted by A335509.
Cf. A056239, A056986, A112798, A158005, A158009, A181796, A335452, A335463.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[Select[Permutations[primeMS[n]],MatchQ[#,{_,x_,_,y_,_,x_,_}/;x

A335471 Number of compositions of n avoiding the pattern (1,2,1).

Original entry on oeis.org

1, 1, 2, 4, 7, 13, 23, 40, 67, 115, 190, 311, 505, 807, 1285, 2031, 3164, 4896, 7550, 11499, 17480, 26379, 39558, 58946, 87469, 129051, 189484, 277143, 403477, 584653, 844236, 1213743, 1738372, 2481770, 3528698, 5003364, 7070225, 9958387, 13982822, 19580613, 27333403

Offset: 0

Gus Wiseman, Jun 17 2020

nonn

Also the number of (1,1,2)-avoiding or (2,1,1)-avoiding compositions.
We define a pattern to be a finite sequence covering an initial interval of positive integers. Patterns are counted by A000670 and ranked by A333217. A sequence S is said to match a pattern P if there is a not necessarily contiguous subsequence of S whose parts have the same relative order as P. For example, (3,1,1,3) matches (1,1,2), (2,1,1), and (2,1,2), but avoids (1,2,1), (1,2,2), and (2,2,1).
A composition of n is a finite sequence of positive integers summing to n.

			The a(0) = 1 through a(5) = 13 compositions:
  ()  (1)  (2)   (3)    (4)     (5)
           (11)  (12)   (13)    (14)
                 (21)   (22)    (23)
                 (111)  (31)    (32)
                        (112)   (41)
                        (211)   (113)
                        (1111)  (122)
                                (212)
                                (221)
                                (311)
                                (1112)
                                (2111)
                                (11111)

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Wikipedia, Permutation pattern
Gus Wiseman, Sequences counting and ranking compositions by the patterns they match or avoid.

The version for patterns is A001710.
The version for prime indices is A335449.
These compositions are ranked by A335467.
The complement A335470 is the matching version.
The (2,1,2)-avoiding version is A335473.
Constant patterns are counted by A000005 and ranked by A272919.
Permutations are counted by A000142 and ranked by A333218.
Patterns are counted by A000670 and ranked by A333217.
Compositions are counted by A011782.
Compositions avoiding (1,2,3) are counted by A102726.
Non-unimodal compositions are counted by A115981 and ranked by A335373.
Combinatory separations are counted by A269134.
Patterns matched by compositions are counted by A335456.
Minimal patterns avoided by a standard composition are counted by A335465.
Cf. A034691, A056986, A106356, A232464, A238279, A333213, A333755.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],!MatchQ[#,{_,x_,_,y_,_,x_,_}/;x

a(n)={local(Cache=Map()); my(F(n,m,k)=if(m>n, m=n); if(m==0, n==0, my(hk=[n,m,k], z); if(!mapisdefined(Cache,hk,&z), z=self()(n,m-1,k) + k*sum(i=1,n\m, self()(n-i*m, m-1, k+i)); mapput(Cache, hk, z)); z)); F(n,n,1)} \\ Andrew Howroyd, Dec 31 2020

a(n > 0) = 2^(n - 1) - A335470(n).

a(n) = F(n,n,1) where F(n,m,k) = F(n,m-1,k) + k*(Sum_{i=1..floor(n/m)} F(n-i*m, m-1, k+i)) for m > 0 with F(0,m,k)=1 and F(n,0,k)=0 otherwise. - Andrew Howroyd, Dec 31 2020

Terms a(21) and beyond from Andrew Howroyd, Dec 31 2020

A335467 Numbers k such that the k-th composition in standard order (A066099) avoids the pattern (1,2,1).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 26, 28, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 46, 47, 48, 50, 52, 56, 58, 60, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 78, 79, 80, 81

Offset: 1

Gus Wiseman, Jun 16 2020

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.

We define a pattern to be a finite sequence covering an initial interval of positive integers. Patterns are counted by A000670 and ranked by A333217. A sequence S is said to match a pattern P if there is a not necessarily contiguous subsequence of S whose parts have the same relative order as P. For example, (3,1,1,3) matches (1,1,2), (2,1,1), and (2,1,2), but avoids (1,2,1), (1,2,2), and (2,2,1).

			See A335466 for an example of the complement.

Wikipedia, Permutation pattern
Gus Wiseman, Sequences counting and ranking compositions by the patterns they match or avoid.
Gus Wiseman, Statistics, classes, and transformations of standard compositions

The complement A335466 is the matching version.
The (2,1,2)-avoiding version is A335469.
These compositions are counted by A335471.
Constant patterns are counted by A000005 and ranked by A272919.
Permutations are counted by A000142 and ranked by A333218.
Patterns are counted by A000670 and ranked by A333217.
Non-unimodal compositions are counted by A115981 and ranked by A335373.
Combinatory separations are counted by A269134 and ranked by A334030.
Patterns matched by standard compositions are counted by A335454.
Minimal patterns avoided by a standard composition are counted by A335465.
Cf. A001710, A034691, A056986, A108917, A114994, A238279, A333224, A333257, A335449, A335456, A335458.

stc[n_]:=Reverse[Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]];
Select[Range[0,100],!MatchQ[stc[#],{_,x_,_,y_,_,x_,_}/;x

A335450 Number of (2,1,2)-avoiding permutations of the prime indices of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 2, 1, 1, 2, 1, 3, 2, 2, 1, 4, 1, 2, 1, 3, 1, 6, 1, 1, 2, 2, 2, 3, 1, 2, 2, 4, 1, 6, 1, 3, 3, 2, 1, 5, 1, 2, 2, 3, 1, 2, 2, 4, 2, 2, 1, 12, 1, 2, 3, 1, 2, 6, 1, 3, 2, 6, 1, 4, 1, 2, 2, 3, 2, 6, 1, 5, 1, 2, 1, 12, 2, 2

Offset: 1

Gus Wiseman, Jun 14 2020

nonn

Depends only on unsorted prime signature (A124010), but not only on sorted prime signature (A118914).
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.
We define a pattern to be a finite sequence covering an initial interval of positive integers. Patterns are counted by A000670. A sequence S is said to match a pattern P if there is a not necessarily contiguous subsequence of S whose parts have the same relative order as P. For example, (3,1,1,3) matches (1,1,2), (2,1,1), and (2,1,2), but avoids (1,2,1), (1,2,2), and (2,2,1).

			The permutations for n = 2, 6, 12, 24, 30, 48, 60, 90:
  (1)  (12)  (112)  (1112)  (123)  (11112)  (1123)  (1223)
       (21)  (211)  (2111)  (132)  (21111)  (1132)  (1322)
                            (213)           (2113)  (2123)
                            (231)           (2311)  (2213)
                            (312)           (3112)  (2231)
                            (321)           (3211)  (3122)
                                                    (3212)
                                                    (3221)

Wikipedia, Permutation pattern
Gus Wiseman, Sequences counting and ranking compositions by the patterns they match or avoid.

Positions of ones are A000961.
Replacing (2,1,2) with (1,2,1) gives A335449.
The matching version is A335453.
Patterns are counted by A000670.
(2,1,2)-avoiding patterns are counted by A001710.
Permutations of prime indices are counted by A008480.
Unsorted prime signature is A124010. Sorted prime signature is A118914.
(1,2,1) and (2,1,2)-avoiding permutations of prime indices are A333175.
STC-numbers of permutations of prime indices are A333221.
(1,2,1) and (2,1,2)-avoiding permutations of prime indices are A335448.
Patterns matched by standard compositions are counted by A335454.
(1,2,1) or (2,1,2)-matching permutations of prime indices are A335460.
(1,2,1) and (2,1,2)-matching permutations of prime indices are A335462.
Dimensions of downsets of standard compositions are A335465.
(2,1,2)-avoiding compositions are ranked by A335469.
(2,1,2)-avoiding compositions are counted by A335473.
(2,2,1)-avoiding compositions are ranked by A335524.
(1,2,2)-avoiding compositions are ranked by A335525.
Cf. A056239, A056986, A112798, A158005, A181796, A335452, A335463.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[Select[Permutations[primeMS[n]],!MatchQ[#,{_,x_,_,y_,_,x_,_}/;x>y]&]],{n,100}]

A335522 Numbers k such that the k-th composition in standard order (A066099) avoids the pattern (1,1,2).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 47, 48, 49, 50, 51, 52, 53, 55, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 79, 80, 81

Offset: 1

Gus Wiseman, Jun 18 2020

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.

We define a pattern to be a finite sequence covering an initial interval of positive integers. Patterns are counted by A000670 and ranked by A333217. A sequence S is said to match a pattern P if there is a not necessarily contiguous subsequence of S whose parts have the same relative order as P. For example, (3,1,1,3) matches (1,1,2), (2,1,1), and (2,1,2), but avoids (1,2,1), (1,2,2), and (2,2,1).

Wikipedia, Permutation pattern
Gus Wiseman, Sequences counting and ranking compositions by the patterns they match or avoid.
Gus Wiseman, Statistics, classes, and transformations of standard compositions

Patterns avoiding this pattern are counted by A001710 (by length).
Permutations of prime indices avoiding this pattern are counted by A335449.
These compositions are counted by A335471 (by sum).
The complement A335476 is the matching version.
The (2,1,1)-avoiding version is A335523.
Constant patterns are counted by A000005 and ranked by A272919.
Permutations are counted by A000142 and ranked by A333218.
Patterns are counted by A000670 and ranked by A333217.
Non-unimodal compositions are counted by A115981 and ranked by A335373.
Combinatory separations are counted by A269134.
Patterns matched by standard compositions are counted by A335454.
Minimal patterns avoided by a standard composition are counted by A335465.
Cf. A034691, A056986, A108917, A114994, A238279, A333224, A333257, A335446, A335456, A335458.

stc[n_]:=Reverse[Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]];
Select[Range[0,100],!MatchQ[stc[#],{_,x_,_,x_,_,y_,_}/;x

A335523 Numbers k such that the k-th composition in standard order (A066099) avoids the pattern (2,1,1).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22, 24, 25, 26, 28, 29, 30, 31, 32, 33, 34, 36, 37, 38, 40, 41, 42, 44, 48, 49, 50, 52, 53, 54, 56, 57, 58, 60, 61, 62, 63, 64, 65, 66, 68, 69, 70, 72, 73, 76, 80, 81, 82, 84, 85, 86, 88, 90

Offset: 1

Gus Wiseman, Jun 18 2020

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.

We define a pattern to be a finite sequence covering an initial interval of positive integers. Patterns are counted by A000670 and ranked by A333217. A sequence S is said to match a pattern P if there is a not necessarily contiguous subsequence of S whose parts have the same relative order as P. For example, (3,1,1,3) matches (1,1,2), (2,1,1), and (2,1,2), but avoids (1,2,1), (1,2,2), and (2,2,1).

Wikipedia, Permutation pattern
Gus Wiseman, Sequences counting and ranking compositions by the patterns they match or avoid.
Gus Wiseman, Statistics, classes, and transformations of standard compositions

Patterns avoiding this pattern are counted by A001710 (by length).
Permutations of prime indices avoiding this pattern are counted by A335449.
These compositions are counted by A335471 (by sum).
The complement A335478 is the matching version.
The (1,1,2)-avoiding version is A335522.
Constant patterns are counted by A000005 and ranked by A272919.
Permutations are counted by A000142 and ranked by A333218.
Patterns are counted by A000670 and ranked by A333217.
Non-unimodal compositions are counted by A115981 and ranked by A335373.
Combinatory separations are counted by A269134.
Patterns matched by standard compositions are counted by A335454.
Minimal patterns avoided by a standard composition are counted by A335465.
Cf. A034691, A056986, A108917, A114994, A238279, A333224, A333257, A335446, A335456, A335458.

stc[n_]:=Reverse[Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]];
Select[Range[0,100],!MatchQ[stc[#],{_,x_,_,y_,_,y_,_}/;x>y]&]

cp's OEIS Frontend

A374768 Numbers k such that the leaders of weakly increasing runs in the k-th composition in standard order (A066099) are distinct.

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A333175 If n = Product (p_j^k_j) then a(n) = Sum (a(n/p_j^k_j)), with a(1) = 1.

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A335446 Number of (1,2,1)-matching permutations of the prime indices of n.

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A335471 Number of compositions of n avoiding the pattern (1,2,1).

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A335467 Numbers k such that the k-th composition in standard order (A066099) avoids the pattern (1,2,1).

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A335450 Number of (2,1,2)-avoiding permutations of the prime indices of n.

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A335522 Numbers k such that the k-th composition in standard order (A066099) avoids the pattern (1,1,2).

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A335523 Numbers k such that the k-th composition in standard order (A066099) avoids the pattern (2,1,1).

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