A269134 -id:A269134 - cp's OEIS frontend

A335525 Numbers k such that the k-th composition in standard order (A066099) avoids the pattern (1,2,2).

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 55, 56, 57, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71

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 A335450.
These compositions are counted by A335473 (by sum).
The complement A335475 is the matching version.
The (2,2,1)-avoiding version is A335524.
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, A334968, 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

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

Original entry on oeis.org

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.

A337505 Number of sequences of length 2*n covering an initial interval of positive integers and splitting into n maximal anti-runs.

Original entry on oeis.org

1, 2, 24, 440, 10780, 329112, 12006456, 508903824, 24559486380, 1328964785720, 79670488601704, 5240336913228144, 375167786246499064, 29038998659140223600, 2416268289647552828400, 215068032231876851531040, 20389611819955706893052460, 2051184695261785540782403320

Offset: 0

Gus Wiseman, Sep 05 2020

nonn

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

			The a(2) = 24 sequences:
  (2,1,2,2)  (1,2,3,3)  (1,2,2,3)  (1,1,2,3)
  (2,2,1,2)  (1,3,3,2)  (1,3,2,2)  (1,1,3,2)
  (1,2,2,1)  (2,1,3,3)  (2,2,1,3)  (2,1,1,3)
  (2,1,1,2)  (2,3,3,1)  (2,2,3,1)  (2,3,1,1)
  (1,1,2,1)  (3,3,1,2)  (3,1,2,2)  (3,1,1,2)
  (1,2,1,1)  (3,3,2,1)  (3,2,2,1)  (3,2,1,1)

Andrew Howroyd, Table of n, a(n) for n = 0..200

A336108 is the version for compositions and runs.
A337504 is the version for compositions.
A337506 has this as main diagonal n = 2*k.
A337564 is the version for runs.
A000670 counts sequences covering an initial interval.
A003242 counts anti-run compositions.
A005649 counts anti-runs covering an initial interval.
A124767 counts maximal runs in standard compositions.
A333381 counts maximal anti-runs in standard compositions.
A333769 gives run-lengths in standard compositions.
A337565 gives anti-run lengths in standard compositions.
Cf. A052841, A106351, A106356, A269134, A325535, A333489, A333627, A333755.

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[2*n],Length[Split[#,UnsameQ]]==n&]],{n,0,3}]

\\ here b(n) is A005649.
b(n) = {sum(k=0, n, stirling(n,k,2)*(k + 1)!)}
a(n) = {b(n)*binomial(2*n-1,n)} \\ Andrew Howroyd, Dec 31 2020

a(n) = A005649(n)*binomial(2*n-1,n). - Andrew Howroyd, Dec 31 2020

Terms a(5) and beyond from Andrew Howroyd, Dec 31 2020

A316219 Number of triangles of weight prime(n) in the multiorder of integer partitions of prime numbers into prime parts.

Original entry on oeis.org

1, 1, 3, 6, 15, 31, 92, 161, 464, 2347, 3987, 18202, 50136, 81722, 214976, 903048, 3684567, 5842249, 23206424, 57341256, 89938662, 343306266, 829972421, 3084219358, 17375700038, 40920517008, 62656899579, 146415515992, 223442878751, 518427758704, 9544240589455, 21746920337606

Offset: 1

Gus Wiseman, Jun 26 2018

nonn

A prime partition is an integer partition of a prime number into prime parts. Then a(n) is the number of sequences of prime partitions whose sums are weakly decreasing and sum to the n-th prime number.

Andrew Howroyd, Table of n, a(n) for n = 1..1000
Gus Wiseman, Comcategories and Multiorders
Gus Wiseman, Illustration of the first five terms of A316219.

Cf. A000040, A000041, A000607, A056768, A063834, A100118, A269134, A281113, A316220.

nn=20;
pen[n_]:=pen[n]=SeriesCoefficient[Product[1/(1-x^p),{p,Select[Range[n],PrimeQ]}],{x,0,n}]
Table[Sum[Times@@pen/@p,{p,Select[IntegerPartitions[Prime[n]],And@@PrimeQ/@#&]}],{n,nn}]

P(n,f)={1/prod(k=1, n, 1 - f(k)*x^prime(k) + O(x*x^prime(n)))}
seq(n)={my(p=P(n, i->1), q=P(n, i->polcoef(p, prime(i)))); vector(n, k, polcoef(q, prime(k)))} \\ Andrew Howroyd, Jan 16 2023

Terms a(16) and beyond from Andrew Howroyd, Jan 16 2023

A318560 Number of combinatory separations of a multiset whose multiplicities are the prime indices of n in weakly decreasing order.

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 5, 3, 8, 7, 7, 8, 11, 12, 15, 5, 15, 17, 22, 14, 27, 19, 30, 13, 27, 30, 33, 26, 42, 37, 56, 7, 44, 45, 51, 34, 77, 67, 72, 25

Offset: 1

Gus Wiseman, Aug 28 2018

A multiset is normal if it spans an initial interval of positive integers. The type of a multiset is the unique normal multiset that has the same sequence of multiplicities when its entries are taken in increasing order. For example the type of 335556 is 112223. A (headless) combinatory separation of a multiset m is a multiset of normal multisets {t_1,...,t_k} such that there exist multisets {s_1,...,s_k} with multiset union m and such that s_i has type t_i for each i = 1...k.

The prime indices of n are the n-th row of A296150.

			The a(18) = 17 combinatory separations of {1,1,2,2,3}:
  {11223}
  {1,1122} {1,1123} {1,1223} {11,112} {12,112} {12,122} {12,123}
  {1,1,112} {1,1,122} {1,1,123} {1,11,11} {1,11,12} {1,12,12}
  {1,1,1,11} {1,1,1,12}
  {1,1,1,1,1}

Cf. A007716, A056239, A255906, A269134, A296150, A305936.

Cf. A317791, A318283, A318284, A318285, A318559, A318562, A318566, A318567.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];
normize[m_]:=m/.Rule@@@Table[{Union[m][[i]],i},{i,Length[Union[m]]}];
Table[Length[Union[Sort/@Map[normize,mps[nrmptn[n]],{2}]]],{n,20}]

A318562 Number of combinatory separations of strongly normal multisets of weight n with strongly normal parts.

Original entry on oeis.org

1, 4, 10, 32, 80, 239, 605, 1670, 4251

Offset: 1

Gus Wiseman, Aug 29 2018

A multiset is normal if it spans an initial interval of positive integers, and strongly normal if in addition it has weakly decreasing multiplicities. The type of a multiset of integers is the unique normal multiset that has the same sequence of multiplicities when its entries are taken in increasing order. For example the type of 335556 is 112223.

A pair h<={g_1,...,g_k} is a combinatory separation iff there exists a multiset partition of h whose multiset of block-types is {g_1,...,g_k}. For example, the (headless) combinatory separations of the multiset 1122 are {1122}, {1,112}, {1,122}, {11,11}, {12,12}, {1,1,11}, {1,1,12}, {1,1,1,1}. This list excludes {12,11} because one cannot partition 1122 into two blocks where one block has two distinct elements and the other block has two equal elements.

			The a(3) = 10 combinatory separations:
  111<={111}
  111<={1,11}
  111<={1,1,1}
  112<={112}
  112<={1,11}
  112<={1,12}
  112<={1,1,1}
  123<={123}
  123<={1,12}
  123<={1,1,1}

Cf. A000041, A007716, A255906, A265947, A269134.

Cf. A317533, A317791, A318396, A318559, A318563, A318564, A318565, A318566.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
strnorm[n_]:=Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n];
normize[m_]:=m/.Rule@@@Table[{Union[m][[i]],i},{i,Length[Union[m]]}];
strnormQ[m_]:=OrderedQ[Length/@Split[m],GreaterEqual];
Table[Length[Select[Union@@Table[{m,Sort[normize/@#]}&/@mps[m],{m,strnorm[n]}],And@@strnormQ/@#[[2]]&]],{n,6}]

A335488 Numbers k such that the k-th composition in standard order (A066099) matches the pattern (1,1).

Original entry on oeis.org

3, 7, 10, 11, 13, 14, 15, 19, 21, 22, 23, 25, 26, 27, 28, 29, 30, 31, 35, 36, 39, 42, 43, 45, 46, 47, 49, 51, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 67, 71, 73, 74, 75, 76, 77, 78, 79, 82, 83, 84, 85, 86, 87, 89, 90, 91, 92, 93, 94, 95, 97, 99, 100, 101

Offset: 1

Gus Wiseman, Jun 18 2020

nonn

These are compositions with some part appearing more than once, or non-strict compositions.
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:
   3: (1,1)
   7: (1,1,1)
  10: (2,2)
  11: (2,1,1)
  13: (1,2,1)
  14: (1,1,2)
  15: (1,1,1,1)
  19: (3,1,1)
  21: (2,2,1)
  22: (2,1,2)
  23: (2,1,1,1)
  25: (1,3,1)
  26: (1,2,2)
  27: (1,2,1,1)
  28: (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 complement A233564 is the avoiding version.
Patterns matching this pattern are counted by A019472 (by length).
Permutations of prime indices matching this pattern are counted by A335487.
These compositions are counted by A261982 (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.
The (1,1,1)-matching case is A335512.
Cf. A034691, A056986, A108917, A114994, 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_,_,x_,_}]&]

A335508 Number of patterns of length n matching the pattern (1,1,1).

Original entry on oeis.org

0, 0, 0, 1, 9, 91, 993, 12013, 160275, 2347141, 37496163, 649660573, 12142311195, 243626199181, 5224710549243, 119294328993853, 2889836999693355, 74037381200415901, 2000383612949821323, 56850708386783835133, 1695491518035158123115, 52949018580275965241821

Offset: 0

Gus Wiseman, Jun 18 2020

nonn

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 a(3) = 1 through a(4) = 9 patterns:
  (1,1,1)  (1,1,1,1)
           (1,1,1,2)
           (1,1,2,1)
           (1,2,1,1)
           (1,2,2,2)
           (2,1,1,1)
           (2,1,2,2)
           (2,2,1,2)
           (2,2,2,1)

Alois P. Heinz, Table of n, a(n) for n = 0..424
Wikipedia, Permutation pattern
Gus Wiseman, Sequences counting and ranking compositions by the patterns they match or avoid.

The complement A080599 is the avoiding version.
Permutations of prime indices matching this pattern are counted by A335510.
Compositions matching this pattern are counted by A335455 and ranked by A335512.
Patterns are counted by A000670 and ranked by A333217.
Patterns matching the pattern (1,1) are counted by A019472.
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.
Patterns matching (1,2,3) are counted by A335515.
Cf. A034691, A056986, A232464, A238279, A292884, A333175, A333755, A335451, A335456, A335457, A335458.
Cf. A276922.

b:= proc(n, k) option remember; `if`(n=0, 1, add(
      b(n-i, k)*binomial(n, i), i=1..min(n, k)))
    end:
a:= n-> b(n$2)-b(n, 2):
seq(a(n), n=0..21);  # Alois P. Heinz, Jan 28 2024

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_,_,x_,_,x_,_}]&]],{n,0,6}]

a(n) = Sum_{k=3..n} A276922(n,k). - Alois P. Heinz, Jan 28 2024

a(n) = A000670(n) - A080599(n). - Andrew Howroyd, Jan 28 2024

a(9)-a(21) from Alois P. Heinz, Jan 28 2024

A335512 Numbers k such that the k-th composition in standard order (A066099) matches the pattern (1,1,1).

Original entry on oeis.org

7, 15, 23, 27, 29, 30, 31, 39, 42, 47, 51, 55, 57, 59, 60, 61, 62, 63, 71, 79, 85, 86, 87, 90, 91, 93, 94, 95, 99, 103, 106, 107, 109, 110, 111, 113, 115, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 135, 143, 151, 155, 157, 158, 159, 167, 170, 171

Offset: 1

Gus Wiseman, Jun 18 2020

nonn

These are compositions with some part appearing more than twice.
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:
   7: (1,1,1)
  15: (1,1,1,1)
  23: (2,1,1,1)
  27: (1,2,1,1)
  29: (1,1,2,1)
  30: (1,1,1,2)
  31: (1,1,1,1,1)
  39: (3,1,1,1)
  42: (2,2,2)
  47: (2,1,1,1,1)
  51: (1,3,1,1)
  55: (1,2,1,1,1)
  57: (1,1,3,1)
  59: (1,1,2,1,1)
  60: (1,1,1,3)

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 complement A335513 is the avoiding version.
Patterns matching this pattern are counted by A335508 (by length).
Permutations of prime indices matching this pattern are counted by A335510.
These compositions are counted by A335455 (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.
The (1,1)-matching version is A335488.
Cf. A034691, A056986, A114994, A238279, A334968, A335456, A335458.

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

A318563 Number of combinatory separations of strongly normal multisets of weight n.

Original entry on oeis.org

1, 4, 10, 33, 85, 272, 730, 2197, 6133

Offset: 1

Gus Wiseman, Aug 29 2018

A multiset is normal if it spans an initial interval of positive integers, and strongly normal if in addition it has weakly decreasing multiplicities. The type of a multiset of integers is the unique normal multiset that has the same sequence of multiplicities when its entries are taken in increasing order. For example the type of 335556 is 112223.

A pair h<={g_1,...,g_k} is a combinatory separation iff there exists a multiset partition of h whose multiset of block-types is {g_1,...,g_k}. For example, the (headless) combinatory separations of the multiset 1122 are {1122}, {1,112}, {1,122}, {11,11}, {12,12}, {1,1,11}, {1,1,12}, {1,1,1,1}. This list excludes {12,11} because one cannot partition 1122 into two blocks where one block has two distinct elements and the other block has two equal elements.

			The a(3) = 10 combinatory separations:
  111<={111}
  111<={1,11}
  111<={1,1,1}
  112<={112}
  112<={1,11}
  112<={1,12}
  112<={1,1,1}
  123<={123}
  123<={1,12}
  123<={1,1,1}

Cf. A000041, A007716, A255906, A265947, A269134.

Cf. A317533, A317791, A318396, A318559, A318562, A318564, A318565, A318566.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
strnorm[n_]:=Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n];
normize[m_]:=m/.Rule@@@Table[{Union[m][[i]],i},{i,Length[Union[m]]}];
Table[Length[Union@@Table[{m,Sort[normize/@#]}&/@mps[m],{m,strnorm[n]}]],{n,7}]

cp's OEIS Frontend

A335525 Numbers k such that the k-th composition in standard order (A066099) avoids the pattern (1,2,2).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A337505 Number of sequences of length 2*n covering an initial interval of positive integers and splitting into n maximal anti-runs.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A316219 Number of triangles of weight prime(n) in the multiorder of integer partitions of prime numbers into prime parts.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A318560 Number of combinatory separations of a multiset whose multiplicities are the prime indices of n in weakly decreasing order.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A318562 Number of combinatory separations of strongly normal multisets of weight n with strongly normal parts.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A335488 Numbers k such that the k-th composition in standard order (A066099) matches the pattern (1,1).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A335508 Number of patterns of length n matching the pattern (1,1,1).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs