A335550 -id:A335550 - cp's OEIS frontend

A335465 Number of minimal normal patterns avoided by the n-th composition in standard order (A066099).

1, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 3, 12, 4, 3, 3, 3, 3, 4, 3, 4, 12, 4, 3, 12, 4, 12, 4, 12, 4, 3, 3, 3, 3, 4, 3, 3, 6, 4, 3, 6, 3, 3, 6, 10, 10, 4, 3, 12, 6, 12, 3, 10, 10, 12, 4, 12, 3, 12, 4, 12, 4, 3, 3, 3, 3, 4, 3, 3, 6

Offset: 0

Gus Wiseman, Jun 20 2020

These patterns comprise the basis of the class of patterns generated by this composition.
We define a (normal) 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 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 bases of classes generated by (), (1), (2,1,1), (3,1,2), (2,1,2,1), and (1,2,1), corresponding to n = 0, 1, 11, 38, 45, 13, are the respective columns below.
  (1)  (1,1)  (1,2)    (1,1)    (1,1,1)    (1,1,1)
       (1,2)  (1,1,1)  (1,2,3)  (1,1,2)    (1,1,2)
       (2,1)  (2,2,1)  (1,3,2)  (1,2,2)    (1,2,2)
              (3,2,1)  (2,1,3)  (1,2,3)    (1,2,3)
                       (2,3,1)  (1,3,2)    (1,3,2)
                       (3,2,1)  (2,1,3)    (2,1,1)
                                (2,3,1)    (2,1,2)
                                (3,1,2)    (2,1,3)
                                (3,2,1)    (2,2,1)
                                (2,2,1,1)  (2,3,1)
                                           (3,1,2)
                                           (3,2,1)

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 matched by standard compositions are counted by A335454.
Patterns matched by compositions of n are counted by A335456(n).
The version for Heinz numbers of partitions is A335550.
Patterns are counted by A000670 and ranked by A333217.
Knapsack compositions are counted by A325676 and ranked by A333223.
The n-th composition has A334299(n) distinct subsequences.
Cf. A056986, A124767, A124770, A124771, A269134, A333218, A333257, A335549.

A335457 Number of normal patterns contiguously matched by compositions of n.

Original entry on oeis.org

1, 2, 5, 12, 31, 80, 196, 486, 1171, 2787, 6564, 15323, 35403, 81251, 185087, 418918, 942525, 2109143, 4695648, 10405694, 22959156

Offset: 0

Gus Wiseman, Jun 23 2020

We define a (normal) 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 a(0) = 1 through a(3) = 12 pairs of a composition with a contiguously matched pattern:
  ()()  (1)()   (2)()     (3)()
        (1)(1)  (11)()    (12)()
                (2)(1)    (21)()
                (11)(1)   (3)(1)
                (11)(11)  (111)()
                          (12)(1)
                          (21)(1)
                          (111)(1)
                          (12)(12)
                          (21)(21)
                          (111)(11)
                          (111)(111)

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

The version for standard compositions is A335458.
The non-contiguous version is A335456.
Patterns are counted by A000670 and ranked by A333217.
The n-th standard composition has A124771(n) contiguous subsequences.
Patterns contiguously matched by prime indices are A335549.
Minimal avoided patterns of prime indices are counted by A335550.
Cf. A000005, A056986, A108917, A124767, A181796, A269134, A333224, A334299.

mstype[q_]:=q/.Table[Union[q][[i]]->i,{i,Length[Union[q]]}];
Table[Sum[Length[Union[mstype/@ReplaceList[cmp,{_,s___,_}:>{s}]]],{cmp,Join@@Permutations/@IntegerPartitions[n]}],{n,0,10}]

a(16)-a(20) from Jinyuan Wang, Jul 08 2020

A335516 Number of normal patterns contiguously matched by the prime indices of n in increasing or decreasing order, counting multiplicity.

Original entry on oeis.org

1, 2, 2, 3, 2, 3, 2, 4, 3, 3, 2, 5, 2, 3, 3, 5, 2, 5, 2, 5, 3, 3, 2, 7, 3, 3, 4, 5, 2, 4, 2, 6, 3, 3, 3, 7, 2, 3, 3, 7, 2, 4, 2, 5, 5, 3, 2, 9, 3, 5, 3, 5, 2, 7, 3, 7, 3, 3, 2, 7, 2, 3, 5, 7, 3, 4, 2, 5, 3, 4, 2, 10, 2, 3, 5, 5, 3, 4, 2, 9, 5, 3, 2, 7, 3, 3, 3

Offset: 1

Gus Wiseman, Jun 26 2020

nonn

First differs from A181796 at a(180) = 9, A181796(180) = 10.
First differs from A335549 at a(90) = 7, A335549(90) = 8.
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 (normal) 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 contiguously match a pattern P if there is a contiguous subsequence of S whose parts have the same relative order as P. For example, (3,1,1,3) contiguously matches (1,1,2) and (2,1,1) but not (2,1,2), (1,2,1), (1,2,2), or (2,2,1).

			The a(n) patterns for n = 2, 30, 12, 60, 120, 540, 1500:
  ()   ()     ()     ()      ()       ()        ()
  (1)  (1)    (1)    (1)     (1)      (1)       (1)
       (12)   (11)   (11)    (11)     (11)      (11)
       (123)  (12)   (12)    (12)     (12)      (12)
              (112)  (112)   (111)    (111)     (111)
                     (123)   (112)    (112)     (112)
                     (1123)  (123)    (122)     (122)
                             (1112)   (1112)    (123)
                             (1123)   (1122)    (1123)
                             (11123)  (1222)    (1222)
                                      (11222)   (1233)
                                      (12223)   (11233)
                                      (112223)  (12333)
                                                (112333)

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

The version for standard compositions is A335458.
The not necessarily contiguous version is A335549.
Patterns are counted by A000670 and ranked by A333217.
A number's prime indices are given in the rows of A112798.
Contiguous subsequences of standard compositions are A124771.
Contiguous sub-partitions of prime indices are counted by A335519.
Minimal avoided patterns of prime indices are counted by A335550.
Patterns contiguously matched by partitions are counted by A335838.
Cf. A000005, A056239, A056986, A108917, A124770, A181796, A269134, A333224, A334299, A335457, A335837.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
mstype[q_]:=q/.Table[Union[q][[i]]->i,{i,Length[Union[q]]}];
Table[Length[Union[mstype/@ReplaceList[primeMS[n],{_,s___,_}:>{s}]]],{n,100}]

A335838 Number of normal patterns contiguously matched by integer partitions of n.

Original entry on oeis.org

1, 2, 5, 9, 18, 31, 54, 89, 145, 225, 349, 524, 778, 1137, 1645, 2330, 3293, 4586, 6341, 8676, 11794, 15880, 21292, 28298, 37419, 49163, 64301, 83576, 108191, 139326, 178699, 228183, 290286, 367760, 464374, 584146, 732481, 915468, 1140773, 1417115, 1755578

Offset: 0

Gus Wiseman, Jun 27 2020

nonn

We define a (normal) 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 contiguously match a pattern P if there is a contiguous subsequence of S whose parts have the same relative order as P. For example, (3,1,1,3) contiguously matches (1,1,2) and (2,1,1) but not (2,1,2), (1,2,1), (1,2,2), or (2,2,1).

			The patterns contiguously matched by (3,2,2,1) are: (), (1), (1,1), (2,1), (2,1,1), (2,2,1), (3,2,2,1). Note that (3,2,1) is not contiguously matched. See A335837 for a larger example.

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

The version for compositions in standard order is A335474.
The version for compositions is A335457.
The not necessarily contiguous version is A335837.
Patterns are counted by A000670 and ranked by A333217.
Patterns contiguously matched by prime indices are counted by A335516.
Contiguous divisors are counted by A335519.
Minimal patterns avoided by prime indices are counted by A335550.
Cf. A056986, A108917, A333257, A335454, A335456, A335458, A335549.

mstype[q_]:=q/.Table[Union[q][[i]]->i,{i,Length[Union[q]]}];
Table[Sum[Length[Union[mstype/@ReplaceList[y,{_,s___,_}:>{s}]]],{y,IntegerPartitions[n]}],{n,0,8}]

More terms from Jinyuan Wang, Jun 27 2020

A335519 Number of contiguous divisors of n.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 4, 3, 4, 2, 6, 2, 4, 4, 5, 2, 6, 2, 6, 4, 4, 2, 8, 3, 4, 4, 6, 2, 7, 2, 6, 4, 4, 4, 9, 2, 4, 4, 8, 2, 7, 2, 6, 6, 4, 2, 10, 3, 6, 4, 6, 2, 8, 4, 8, 4, 4, 2, 10, 2, 4, 6, 7, 4, 7, 2, 6, 4, 7, 2, 12, 2, 4, 6, 6, 4, 7, 2, 10, 5, 4, 2, 10, 4

Offset: 1

Gus Wiseman, Jun 26 2020

nonn

A divisor of n is contiguous if its prime factors, counting multiplicity, are a contiguous subsequence of the prime factors of n. Explicitly, a divisor d|n is contiguous if n can be written as n = x * d * y where the least prime factor of d is at least the greatest prime factor of x, and the greatest prime factor of d is at most the least prime factor of y.

			The a(84) = 10 distinct contiguous subsequences of (2,2,3,7) are (), (2), (3), (7), (2,2), (2,3), (3,7), (2,2,3), (2,3,7), (2,2,3,7), corresponding to the divisors 1, 2, 3, 7, 4, 6, 21, 12, 42, 84.

Wikipedia, Permutation pattern
Gus Wiseman, Sequences counting and encoding certain classes of multisets
Gus Wiseman, Sequences counting and ranking compositions by the patterns they match or avoid.

The not necessarily contiguous version is A000005.
Each number's prime indices are given in the rows of A112798.
Contiguous subsequences of standard compositions are counted by A124771.
Minimal avoided patterns of prime indices are counted by A335550.
Patterns contiguously matched by partitions are counted by A335838.
Cf. A000670, A056239, A056986, A181796, A325770, A334299, A335457.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[Union[ReplaceList[primeMS[n],{_,s___,_}:>{s}]]],{n,100}]

a(n) = A325770(n) + 1.

A335837 Number of normal patterns matched by integer partitions of n.

Original entry on oeis.org

1, 2, 5, 9, 18, 31, 54, 89, 146, 228, 358, 545, 821, 1219, 1795, 2596, 3741, 5323, 7521, 10534, 14659, 20232, 27788, 37897, 51410, 69347, 93111, 124348, 165378, 218924, 288646, 379021, 495864, 646272, 839490, 1086693, 1402268, 1803786, 2313498, 2958530, 3773093

Offset: 0

Gus Wiseman, Jun 27 2020

nonn

We define a (normal) 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 a(0) = 1 through a(4) = 18  pairs of a partition with a matched pattern:
  ()/()  (1)/()   (2)/()     (3)/()       (4)/()
         (1)/(1)  (2)/(1)    (3)/(1)      (4)/(1)
                  (11)/()    (21)/()      (31)/()
                  (11)/(1)   (21)/(1)     (31)/(1)
                  (11)/(11)  (21)/(21)    (31)/(21)
                             (111)/()     (22)/()
                             (111)/(1)    (22)/(1)
                             (111)/(11)   (22)/(11)
                             (111)/(111)  (211)/()
                                          (211)/(1)
                                          (211)/(11)
                                          (211)/(21)
                                          (211)/(211)
                                          (1111)/()
                                          (1111)/(1)
                                          (1111)/(11)
                                          (1111)/(111)
                                          (1111)/(1111)

Christian Sievers, Table of n, a(n) for n = 0..2000
Wikipedia, Permutation pattern
Gus Wiseman, Sequences counting and ranking compositions by the patterns they match or avoid.

The version for compositions in standard order is A335454.
The version for compositions is A335456.
The version for Heinz numbers of partitions is A335549.
The contiguous case is A335838.
Patterns are counted by A000670 and ranked by A333217.
Patterns contiguously matched by prime indices are A335516.
Contiguous divisors are counted by A335519.
Minimal patterns avoided by prime indices are counted by A335550.
Cf. A000005, A056986, A108917, A269134, A333257, A334299, A335458, A335465.

mstype[q_]:=q/.Table[Union[q][[i]]->i,{i,Length[Union[q]]}];
Table[Sum[Length[Union[mstype/@Subsets[y]]],{y,IntegerPartitions[n]}],{n,0,8}]

lista(n) = {
  my(v=vector(n+1,i,1+x*O(x^n)));
  for(k=1,n,
    v=vector(n\(k+1)+1,i,
        (1-x^(i*k))/(1-x^k)*v[i] + sum(j=i,n\k,x^(j*k)*v[j+1]) +
        x^(k*i)/(1-x^k)^2*v[1] ) );
  Vec(v[1]) } \\ Christian Sievers, May 08 2025

a(18) corrected by and a(19)-a(22) from Jinyuan Wang, Jun 27 2020

More terms from Christian Sievers, May 08 2025

cp's OEIS Frontend

A335465 Number of minimal normal patterns avoided by the n-th composition in standard order (A066099).

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A335457 Number of normal patterns contiguously matched by compositions of n.

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A335516 Number of normal patterns contiguously matched by the prime indices of n in increasing or decreasing order, counting multiplicity.

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A335838 Number of normal patterns contiguously matched by integer partitions of n.

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A335519 Number of contiguous divisors of n.

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A335837 Number of normal patterns matched by integer partitions of n.

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