A335516 -id:A335516 - cp's OEIS frontend

A335550 Number of minimal normal patterns avoided by the prime indices of n in increasing or decreasing order, counting multiplicity.

1, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 3, 3, 3, 3, 3, 4, 3, 4, 3, 3, 3, 4, 3, 3, 3, 4, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 3, 3, 3, 4, 4, 3, 3, 4, 3, 4, 3, 4, 3, 4, 3, 4, 3, 3, 3, 4, 3, 3, 4, 3, 3, 3, 3, 4, 3, 3

Offset: 1

Gus Wiseman, Jun 26 2020

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 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(12) = 4 minimal patterns avoiding (1,1,2) are: (2,1), (1,1,1), (1,2,2), (1,2,3).
The a(30) = 3 minimal patterns avoiding (1,2,3) are: (1,1), (2,1), (1,2,3,4).

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

The version for standard compositions is A335465.
Patterns are counted by A000670.
Sum of prime indices is A056239.
Each number's prime indices are given in the rows of A112798.
Patterns are ranked by A333217.
Patterns matched by compositions are counted by A335456.
Patterns matched by prime indices are counted by A335549.
Patterns matched by partitions are counted by A335837.
Cf. A124770, A124771, A181796, A269134, A299702, A333257, A335452, A335516.

It appears that for n > 1, a(n) = 3 if n is a power of a squarefree number (A072774), and a(n) = 4 otherwise.

A232580 Number of binary sequences of length n that contain at least one contiguous subsequence 011.

Original entry on oeis.org

0, 0, 0, 1, 4, 12, 31, 74, 168, 369, 792, 1672, 3487, 7206, 14788, 30185, 61356, 124308, 251199, 506578, 1019920, 2050785, 4119280, 8267216, 16580799, 33236622, 66594636, 133385689, 267089188, 534692604, 1070217247, 2141780762, 4285739832, 8575004241

Offset: 0

Geoffrey Critzer, Nov 26 2013

From Gus Wiseman, Jun 26 2022: (Start)
Also the number of integer compositions of n + 1 with an even part other than the first or last. For example, the a(3) = 1 through a(5) = 12 compositions are:
  (121)  (122)   (123)
         (221)   (141)
         (1121)  (222)
         (1211)  (321)
                 (1122)
                 (1212)
                 (1221)
                 (2121)
                 (2211)
                 (11121)
                 (11211)
                 (12111)
The odd version is A274230.
(End)

			a(4) = 4 because we have: 0011, 0110, 0111, 1011.

Colin Barker, Table of n, a(n) for n = 0..1000
Philippe Flajolet and Robert Sedgewick, Analytic Combinatorics, Cambridge Univ. Press, 2009, page 34.
Index entries for linear recurrences with constant coefficients, signature (4,-4,-1,2).

The complement is counted by A000071(n) = A001911(n) + 1.
For the contiguous pattern (1,1) or (0,0) we have A000225.
For the contiguous pattern (1,0,1) or (0,1,0) we have A000253.
For the contiguous pattern (1,0) or (0,1) we have A000295.
Numbers whose binary expansion is of this type are A004750.
For the contiguous pattern (1,1,1) or (0,0,0) we have A050231.
The not necessarily contiguous version is A324172.
Cf. A034691, A261983, A274230, A335455, A335457, A335458, A335516.

nn=40;a=x/(1-x);CoefficientList[Series[a^2 x/(1-a x)/(1-2x),{x,0,nn}],x]
(* second program *)
Table[Length[Select[Tuples[{0,1},n],MatchQ[#,{_,0,1,1,_}]&]],{n,0,10}] (* Gus Wiseman, Jun 26 2022 *)

concat(vector(3), Vec(x^3/(-2*x^4+x^3+4*x^2-4*x+1) + O(x^40))) \\ Colin Barker, Nov 03 2016

O.g.f.: x^3/( (1-x)^2*(1-x^2/(1-x))*(1-2x) ).
a(n) ~ 2^n.
From Colin Barker, Nov 03 2016: (Start)
a(n) = (1 + 2^n - (2^(-n)*((1-sqrt(5))^n*(-2+sqrt(5)) + (1+sqrt(5))^n*(2+sqrt(5))))/sqrt(5)).
a(n) = 4*a(n-1) - 4*a(n-2) - a(n-3) + 2*a(n-4) for n > 3. (End)
a(n) = 2^n - Fibonacci(n+3) + 1. - Ehren Metcalfe, Dec 27 2018
E.g.f.: 2*exp(x/2)*(5*exp(x)*cosh(x/2) - 5*cosh(sqrt(5)*x/2) - 2*sqrt(5)*sinh(sqrt(5)*x/2))/5. - Stefano Spezia, Apr 06 2022

A335474 Number of nonempty normal patterns contiguously matched by the n-th composition in standard order.

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 2, 3, 1, 2, 2, 4, 2, 4, 4, 4, 1, 2, 2, 4, 2, 4, 4, 6, 2, 4, 4, 7, 4, 7, 6, 5, 1, 2, 2, 4, 2, 3, 4, 6, 2, 4, 3, 6, 4, 6, 7, 8, 2, 4, 4, 7, 3, 7, 6, 10, 4, 7, 6, 10, 6, 10, 8, 6, 1, 2, 2, 4, 2, 3, 4, 6, 2, 4, 4, 6, 4, 6, 7, 8, 2, 4, 4, 7, 4, 6

Offset: 0

Gus Wiseman, Jun 21 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 (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(n) patterns for n = 32, 80, 133, 290, 305, 329, 436 are:
      (1)  (1)   (1)    (1)    (1)     (1)     (1)
           (12)  (21)   (12)   (12)    (11)    (12)
                 (321)  (21)   (21)    (12)    (21)
                        (231)  (121)   (21)    (121)
                               (213)   (122)   (123)
                               (2131)  (221)   (212)
                                       (2331)  (1212)
                                               (2123)
                                               (12123)

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 for Heinz numbers of partitions is A335516(n) - 1.
The non-contiguous version is A335454(n) - 1.
The version allowing empty patterns is A335458.
Patterns are counted by A000670 and ranked by A333217.
The n-th composition has A124771(n) distinct consecutive subsequences.
Knapsack compositions are counted by A325676 and ranked by A333223.
The n-th composition has A334299(n) distinct subsequences.
Minimal avoided patterns are counted by A335465.
Patterns matched by prime indices are counted by A335549.
Cf. A034691, A056986, A108917, A124767, A124770, A181796, A269134, A333222, A333224, A335456, A335457.

stc[n_]:=Reverse[Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]];
mstype[q_]:=q/.Table[Union[q][[i]]->i,{i,Length[Union[q]]}];
Table[Length[Union[mstype/@ReplaceList[stc[n],{_,s__,_}:>{s}]]],{n,0,100}]

a(n) = A335458(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

A386576 Number of anti-runs of length n covering an initial interval of positive integers with strictly decreasing multiplicities.

Original entry on oeis.org

1, 1, 0, 1, 0, 1, 10, 4, 14, 84, 1136, 967, 3342, 12823, 101762, 1769580

Offset: 0

Gus Wiseman, Aug 03 2025

An anti-run is a sequence with no adjacent equal terms.

			The a(7) = 4 anti-runs are:
  (1,2,1,2,1,2,1)
  (1,2,1,2,1,3,1)
  (1,2,1,3,1,2,1)
  (1,3,1,2,1,2,1)

For any multiplicities we have A005649.
For weakly instead of strictly decreasing multiplicities we have A321688.
A003242 and A335452 count anti-runs, ranks A333489.
A005651 counts ordered set partitions with weakly decreasing sizes, strict A007837.
A032020 counts strict anti-run compositions.
A325534 counts separable multisets, ranks A335433.
A325535 counts inseparable multisets, ranks A335448.
A336103 counts normal separable multisets, inseparable A336102.
A386583 counts separable partitions by length, inseparable A386584.
A386585 counts partitions of separable type by length, sums A336106, ranks A335127.
A386586 counts partitions of inseparable type by length, sums A025065, ranks A335126.
A386633 counts separable set partitions, row sums of A386635.
A386634 counts inseparable set partitions, row sums of A386636.
Cf. A000670, A019472, A106351, A238130, A335125, A335516, A386580, A386581.

seps[ptn_,fir_]:=If[Total[ptn]==1,{{fir}},Join@@Table[Prepend[#,fir]&/@seps[MapAt[#-1&,ptn,fir],nex],{nex,Select[DeleteCases[Range[Length[ptn]],fir],ptn[[#]]>0&]}]];
seps[ptn_]:=If[Total[ptn]==0,{{}},Join@@(seps[ptn,#]&/@Range[Length[ptn]])];
Table[Sum[Length[seps[y]],{y,Select[IntegerPartitions[n],UnsameQ@@#&]}],{n,0,10}]

A335478 Numbers k such that the k-th composition in standard order (A066099) matches the pattern (2,1,1).

Original entry on oeis.org

11, 19, 23, 27, 35, 39, 43, 45, 46, 47, 51, 55, 59, 67, 71, 74, 75, 77, 78, 79, 83, 87, 89, 91, 92, 93, 94, 95, 99, 103, 107, 109, 110, 111, 115, 119, 123, 131, 135, 138, 139, 141, 142, 143, 147, 149, 150, 151, 153, 154, 155, 156, 157, 158, 159, 163, 167, 171

Offset: 1

Gus Wiseman, Jun 18 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).

			The sequence of terms together with the corresponding compositions begins:
  11: (2,1,1)
  19: (3,1,1)
  23: (2,1,1,1)
  27: (1,2,1,1)
  35: (4,1,1)
  39: (3,1,1,1)
  43: (2,2,1,1)
  45: (2,1,2,1)
  46: (2,1,1,2)
  47: (2,1,1,1,1)
  51: (1,3,1,1)
  55: (1,2,1,1,1)
  59: (1,1,2,1,1)
  67: (5,1,1)
  71: (4,1,1,1)

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 complement A335523 is the avoiding version.
The (1,1,2)-matching version is A335476.
Patterns matching this pattern are counted by A335509 (by length).
Permutations of prime indices matching this pattern are counted by A335516.
These compositions are counted by A335470 (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.
Cf. A034691, A056986, A108917, A114994, A238279, A333224, A333257, A335446, A335456, A335458, A335475.

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

A335550 Number of minimal normal patterns avoided by the prime indices of n in increasing or decreasing order, counting multiplicity.

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A232580 Number of binary sequences of length n that contain at least one contiguous subsequence 011.

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A335474 Number of nonempty normal patterns contiguously matched by the n-th composition in standard order.

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

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A386576 Number of anti-runs of length n covering an initial interval of positive integers with strictly decreasing multiplicities.

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A335478 Numbers k such that the k-th composition in standard order (A066099) matches the pattern (2,1,1).

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