A335465 -id:A335465 - cp's OEIS frontend

A335463 Numbers k such that there exists a permutation of the prime indices of k matching both (1,2,1) and (2,1,2).

36, 72, 90, 100, 108, 126, 144, 180, 196, 198, 200, 216, 225, 234, 252, 270, 288, 300, 306, 324, 342, 350, 360, 378, 392, 396, 400, 414, 432, 441, 450, 468, 484, 500, 504, 522, 525, 540, 550, 558, 576, 588, 594, 600, 612, 630, 648, 650, 666, 675, 676, 684, 700

Offset: 1

Gus Wiseman, Jun 20 2020

nonn

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

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).

			The sequence of terms together with their prime indices begins:
   36: {1,1,2,2}
   72: {1,1,1,2,2}
   90: {1,2,2,3}
  100: {1,1,3,3}
  108: {1,1,2,2,2}
  126: {1,2,2,4}
  144: {1,1,1,1,2,2}
  180: {1,1,2,2,3}
  196: {1,1,4,4}
  198: {1,2,2,5}
  200: {1,1,1,3,3}
  216: {1,1,1,2,2,2}
  225: {2,2,3,3}
  234: {1,2,2,6}
  252: {1,1,2,2,4}
  270: {1,2,2,2,3}
  288: {1,1,1,1,1,2,2}
  300: {1,1,2,3,3}

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

Replacing "and" with "or" gives A126706.
Positions of nonzero terms in A335462.
Permutations of prime indices are counted by A008480.
Unsorted prime signature is A124010. Sorted prime signature is A118914.
STC-numbers of permutations of prime indices are A333221.
Patterns matched by standard compositions are counted by A335454.
Cf. A056239, A056986, A112798, A158005, A181796, A333175, A335451, A335452, A335460, A335465.

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

A335489 Number of strict permutations of the prime indices of n.

Original entry on oeis.org

1, 1, 1, 0, 1, 2, 1, 0, 0, 2, 1, 0, 1, 2, 2, 0, 1, 0, 1, 0, 2, 2, 1, 0, 0, 2, 0, 0, 1, 6, 1, 0, 2, 2, 2, 0, 1, 2, 2, 0, 1, 6, 1, 0, 0, 2, 1, 0, 0, 0, 2, 0, 1, 0, 2, 0, 2, 2, 1, 0, 1, 2, 0, 0, 2, 6, 1, 0, 2, 6, 1, 0, 1, 2, 0, 0, 2, 6, 1, 0, 0, 2, 1, 0, 2, 2, 2

Offset: 1

Gus Wiseman, Jun 19 2020

nonn

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.

Also the number of (1,1)-avoiding permutations of the prime indices of n.

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

Positions of first appearances are A002110 with 2 replaced by 4.
Permutations of prime indices are counted by A008480.
The contiguous version is A335451.
Anti-run permutations of prime indices are counted by A335452.
(1,1,1)-avoiding permutations of prime indices are counted by A335511.
Cf. A056239, A056986, A106356, A112798, A238279, A281188, A333221, A335456, A335460, A335462, A335465.

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

If n is squarefree, a(n) = A001221(n)!; otherwise a(n) = 0.

a(n != 4) = A281188(n); a(4) = 0.

A226316 Expansion of g.f. 1/2 + 1/(1+sqrt(1-8x+8x^2)).

Original entry on oeis.org

1, 1, 3, 12, 56, 284, 1516, 8384, 47600, 275808, 1624352, 9694912, 58510912, 356467392, 2189331648, 13540880384, 84265071360, 527232146944, 3314742364672, 20930141861888, 132673039491072, 843959152564224, 5385800362473472, 34470606645280768, 221213787774230528, 1423139139514138624

Offset: 0

N. J. A. Sloane, Jun 09 2013

nonn

From Robert A. Proctor, Jul 18 2017: (Start)
a(n) is the number of words of length n on {1,2,...,r} with positive multiplicities as 1 <= r <= n avoiding the pattern 123. [This is easy to see from the next comment.]
a(n) is the number of 123-avoiding ordered set partitions of {1,2,...,n}. [This is Cor. 2.3 of the Chen-Dai-Zhou reference.] (End)

			From _Gus Wiseman_, Jun 25 2020: (Start)
The a(0) = 1 through a(3) = 12 words that are (1,2,3)-avoiding and cover an initial interval:
  ()  (1)  (1,1)  (1,1,1)
           (1,2)  (1,1,2)
           (2,1)  (1,2,1)
                  (1,2,2)
                  (1,3,2)
                  (2,1,1)
                  (2,1,2)
                  (2,1,3)
                  (2,2,1)
                  (2,3,1)
                  (3,1,2)
                  (3,2,1)
(End)

Vincenzo Librandi, Table of n, a(n) for n = 0..100
Daniel Birmajer, Juan B. Gil, David S. Kenepp, and Michael D. Weiner, Restricted generating trees for weak orderings, arXiv:2108.04302 [math.CO], 2021.
W. Y. C. Chen, A. Y. L. Dai and R. D. P. Zhou, Ordered Partitions Avoiding a Permutation of Length 3, arXiv preprint arXiv:1304.3187 [math.CO], 2013.
Anders Claesson, Giulio Cerbai, Dana C. Ernst, and Hannah Golab, Pattern-avoiding Cayley permutations via combinatorial species, arXiv:2407.19583 [math.CO], 2024.
Robert A. Proctor and Matthew J. Willis, Parabolic Catalan numbers count flagged Schur functions and their appearances as type A Demazure characters (key polynomials), arXiv preprint arXiv:1706.04649 [math.CO], 2017.
Wikipedia, Permutation pattern
Gus Wiseman, Sequences counting and ranking compositions by the patterns they match or avoid.

Cf. A220097.
Sequences covering an initial interval are counted by A000670.
(1,2,3)-matching permutations are counted by A056986.
(1,2,3)-avoiding permutations are counted by A000108.
(1,2,3)-matching compositions are counted by A335514.
(1,2,3)-avoiding compositions are counted by A102726.
(1,2,3)-matching patterns are counted by A335515.
(1,2,3)-avoiding patterns are counted by A226316 (this sequence).
(1,2,3)-matching permutations of prime indices are counted by A335520.
(1,2,3)-avoiding permutations of prime indices are counted by A335521.
(1,2,3)-matching compositions are ranked by A335479.
Cf. A158005, A158009, A333217, A335465.

a:= proc(n) option remember; `if`(n<4, [1$2, 3, 12][n+1],
      ((9*n-3)*a(n-1) -(16*n-20)*a(n-2) +(8*n-16)*a(n-3))/(n+1))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Jun 18 2013

CoefficientList[Series[1/2 + 1 / (1 + Sqrt[1 - 8 x + 8 x^2]), {x, 0, 30}], x] (* Vincenzo Librandi, Jun 18 2013 *)
allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
Table[Length[Select[Join@@Permutations/@allnorm[n],!MatchQ[#,{_,x_,_,y_,_,z_,_}/;xGus Wiseman, Jun 25 2020 *)

a(n) ~ sqrt((sqrt(2)-1)/Pi)*2^(n-1/2)*(2+sqrt(2))^n/n^(3/2). - Vaclav Kotesovec, Jun 29 2013
Conjecture: (n+1)*a(n) +3*(-3*n+1)*a(n-1) +4*(4*n-5)*a(n-2) +8*(-n+2)*a(n-3)=0. - R. J. Mathar, Apr 02 2015
a(n) = A000670(n) - A335515(n). - Gus Wiseman, Jun 25 2020

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

A335479 Numbers k such that the k-th composition in standard order (A066099) matches the pattern (1,2,3).

Original entry on oeis.org

52, 104, 105, 108, 116, 180, 200, 208, 209, 210, 211, 212, 216, 217, 220, 232, 233, 236, 244, 308, 328, 360, 361, 364, 372, 400, 401, 404, 408, 416, 417, 418, 419, 420, 421, 422, 423, 424, 425, 428, 432, 433, 434, 435, 436, 440, 441, 444, 456, 464, 465, 466

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).

			The sequence of terms together with the corresponding compositions begins:
   52: (1,2,3)
  104: (1,2,4)
  105: (1,2,3,1)
  108: (1,2,1,3)
  116: (1,1,2,3)
  180: (2,1,2,3)
  200: (1,3,4)
  208: (1,2,5)
  209: (1,2,4,1)
  210: (1,2,3,2)
  211: (1,2,3,1,1)
  212: (1,2,2,3)
  216: (1,2,1,4)
  217: (1,2,1,3,1)
  220: (1,2,1,1,3)

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 version counting permutations is A056986.
Patterns matching this pattern are counted by A335515 (by length).
Permutations of prime indices matching this pattern are counted by A335520.
These compositions are counted by A335514 (by sum).
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.
Other permutations:
- A335479 (1,2,3)
- A335480 (1,3,2)
- A335481 (2,1,3)
- A335482 (2,3,1)
- A335483 (3,1,2)
- A335484 (3,2,1)
Cf. A034691, A056986, A108917, A114994, A158005, A238279, A333224, A333257, A334968, A335456, A335458.

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

A335470 Number of compositions of n matching the pattern (1,2,1).

Original entry on oeis.org

0, 0, 0, 0, 1, 3, 9, 24, 61, 141, 322, 713, 1543, 3289, 6907, 14353, 29604, 60640, 123522, 250645, 506808, 1022197, 2057594, 4135358, 8301139, 16648165, 33364948, 66831721, 133814251, 267850803, 536026676, 1072528081, 2145745276, 4292485526, 8586405894, 17174865820

Offset: 0

Gus Wiseman, Jun 17 2020

nonn

Also the number of (1,1,2)-matching or (2,1,1)-matching 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(4) = 1 through a(6) = 9 compositions:
  (121)  (131)   (141)
         (1121)  (1131)
         (1211)  (1212)
                 (1221)
                 (1311)
                 (2121)
                 (11121)
                 (11211)
                 (12111)

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 prime indices is A335446.
These compositions are ranked by A335466.
The complement A335471 is the avoiding version.
The (2,1,2)-matching version is A335472.
The version for patterns is A335509.
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.
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.
Compositions matching (1,2,3) are counted by A335514.
Cf. A261982, A034691, A056986, A106356, A238279, A333755, A335454.

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

a(n > 0) = 2^(n - 1) - A335471(n).

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

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

A335473 Number of compositions of n avoiding the pattern (2,1,2).

Original entry on oeis.org

1, 1, 2, 4, 8, 15, 29, 55, 103, 190, 347, 630, 1134, 2028, 3585, 6291, 10950, 18944, 32574, 55692, 94618, 159758, 268147, 447502, 743097, 1227910, 2020110, 3308302, 5394617, 8757108, 14155386, 22784542, 36529813, 58343498, 92850871, 147254007, 232750871, 366671436

Offset: 0

Gus Wiseman, Jun 17 2020

nonn

Also the number of (1,2,2) or (2,2,1)-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) = 15 compositions:
  ()  (1)  (2)   (3)    (4)     (5)
           (11)  (12)   (13)    (14)
                 (21)   (22)    (23)
                 (111)  (31)    (32)
                        (112)   (41)
                        (121)   (113)
                        (211)   (122)
                        (1111)  (131)
                                (221)
                                (311)
                                (1112)
                                (1121)
                                (1211)
                                (2111)
                                (11111)

Andrew Howroyd, Table of n, a(n) for n = 0..200
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 A335450.
These compositions are ranked by A335469.
The (1,2,1)-avoiding version is A335471.
The complement A335472 is the matching version.
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. A261982, A034691, A056986, A106356, A232464, A238279, A333755.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],!MatchQ[#,{_,x_,_,y_,_,x_,_}/;x>y]&]],{n,0,10}]

a(n)={local(Cache=Map()); my(F(n,m,k) = if(m>n, 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,1,1)} \\ Andrew Howroyd, Dec 31 2020

a(n > 0) = 2^(n - 1) - A335472(n).

a(n) = F(n,1,1) 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 <= n with F(0,m,k)=1 and F(n,m,k)=0 otherwise. - Andrew Howroyd, Dec 31 2020

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

A335480 Numbers k such that the k-th composition in standard order (A066099) matches the pattern (1,3,2).

Original entry on oeis.org

50, 98, 101, 102, 114, 178, 194, 196, 197, 198, 202, 203, 205, 206, 210, 226, 229, 230, 242, 306, 324, 354, 357, 358, 370, 386, 388, 389, 390, 393, 394, 395, 396, 397, 398, 402, 404, 405, 406, 407, 410, 411, 413, 414, 418, 421, 422, 434, 450, 452, 453, 454

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).

			The sequence of terms together with the corresponding compositions begins:
   50: (1,3,2)
   98: (1,4,2)
  101: (1,3,2,1)
  102: (1,3,1,2)
  114: (1,1,3,2)
  178: (2,1,3,2)
  194: (1,5,2)
  196: (1,4,3)
  197: (1,4,2,1)
  198: (1,4,1,2)
  202: (1,3,2,2)
  203: (1,3,2,1,1)
  205: (1,3,1,2,1)
  206: (1,3,1,1,2)
  210: (1,2,3,2)

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 version counting permutations is A056986.
Patterns matching this pattern are counted by A335515 (by length).
Permutations of prime indices matching this pattern are counted by A335520.
These compositions are counted by A335514 (by sum).
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.
Permutations matching (1,3,2,4) are counted by A158009.
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.
Other permutations:
- A335479 (1,2,3)
- A335480 (1,3,2)
- A335481 (2,1,3)
- A335482 (2,3,1)
- A335483 (3,1,2)
- A335484 (3,2,1)
Cf. A034691, A056986, A108917, A114994, A158005, A238279, A333224, A333257, A334968, A335456, A335458.

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

A335482 Numbers k such that the k-th composition in standard order (A066099) matches the pattern (2,3,1).

Original entry on oeis.org

41, 81, 83, 89, 105, 145, 161, 163, 165, 166, 167, 169, 177, 179, 185, 209, 211, 217, 233, 289, 290, 291, 297, 305, 321, 323, 325, 326, 327, 329, 331, 332, 333, 334, 335, 337, 339, 345, 353, 355, 357, 358, 359, 361, 369, 371, 377, 401, 417, 419, 421, 422, 423

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).

			The sequence of terms together with the corresponding compositions begins:
   41: (2,3,1)
   81: (2,4,1)
   83: (2,3,1,1)
   89: (2,1,3,1)
  105: (1,2,3,1)
  145: (3,4,1)
  161: (2,5,1)
  163: (2,4,1,1)
  165: (2,3,2,1)
  166: (2,3,1,2)
  167: (2,3,1,1,1)
  169: (2,2,3,1)
  177: (2,1,4,1)
  179: (2,1,3,1,1)
  185: (2,1,1,3,1)

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

The version counting permutations is A056986.
Patterns matching this pattern are counted by A335515 (by length).
Permutations of prime indices matching this pattern are counted by A335520.
These compositions are counted by A335514 (by sum).
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.
Permutations matching (1,3,2,4) are counted by A158009.
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.
Other permutations:
- A335479 (1,2,3)
- A335480 (1,3,2)
- A335481 (2,1,3)
- A335482 (2,3,1)
- A335483 (3,1,2)
- A335484 (3,2,1)
Cf. A034691, A056986, A108917, A114994, A158005, A238279, A333224, A333257, A334968, A335456, A335458.

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

cp's OEIS Frontend

A335463 Numbers k such that there exists a permutation of the prime indices of k matching both (1,2,1) and (2,1,2).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A335489 Number of strict permutations of the prime indices of n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A226316 Expansion of g.f. 1/2 + 1/(1+sqrt(1-8*x+8*x^2)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A335479 Numbers k such that the k-th composition in standard order (A066099) matches the pattern (1,2,3).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A335470 Number of compositions of n matching the pattern (1,2,1).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A335473 Number of compositions of n avoiding the pattern (2,1,2).

Views

A226316 Expansion of g.f. 1/2 + 1/(1+sqrt(1-8x+8x^2)).