A335511 -id:A335511 - cp's OEIS frontend

A080599 Expansion of e.g.f.: 2/(2-2*x-x^2).

1, 1, 3, 12, 66, 450, 3690, 35280, 385560, 4740120, 64751400, 972972000, 15949256400, 283232149200, 5416632421200, 110988861984000, 2425817682288000, 56333385828720000, 1385151050307024000, 35950878932544576000, 982196278209226080000, 28175806418228108640000

Offset: 0

Detlef Pauly (dettodet(AT)yahoo.de), Feb 24 2003

Number of ordered partitions of {1,..,n} with at most 2 elements per block. - Bob Proctor, Apr 18 2005
In other words, number of preferential arrangements of n things (see A000670) in which each clump has size 1 or 2. - N. J. A. Sloane, Apr 13 2014
Recurrences (of the hypergeometric type of the Jovovic formula) mean: multiplying the sequence vector from the left with the associated matrix of the recurrence coefficients (here: an infinite lower triangular matrix with the natural numbers in the main diagonal and the triangular series in the subdiagonal) recovers the sequence up to an index shift. In that sense, this sequence here and many other sequences of the OEIS are eigensequences. - Gary W. Adamson, Feb 14 2011
Number of intervals in the weak (Bruhat) order of S_n that are Boolean algebras. - Richard Stanley, May 09 2011
a(n) = D^n(1/(1-x)) evaluated at x = 0, where D is the operator sqrt(1+2*x)*d/dx. Cf. A000085, A005442 and A052585. - Peter Bala, Dec 07 2011
From Gus Wiseman, Jul 04 2020: (Start)
Also the number of (1,1,1)-avoiding or cubefree sequences of length n covering an initial interval of positive integers. For example, the a(0) = 1 through a(3) = 12 sequences are:
  ()  (1)  (11)  (112)
           (12)  (121)
           (21)  (122)
                 (123)
                 (132)
                 (211)
                 (212)
                 (213)
                 (221)
                 (231)
                 (312)
                 (321)
(End)

			From _Gus Wiseman_, Jul 04 2020: (Start)
The a(0) = 1 through a(3) = 12 ordered set partitions with block sizes <= 2 are:
  {}  {{1}}  {{1,2}}    {{1},{2,3}}
             {{1},{2}}  {{1,2},{3}}
             {{2},{1}}  {{1,3},{2}}
                        {{2},{1,3}}
                        {{2,3},{1}}
                        {{3},{1,2}}
                        {{1},{2},{3}}
                        {{1},{3},{2}}
                        {{2},{1},{3}}
                        {{2},{3},{1}}
                        {{3},{1},{2}}
                        {{3},{2},{1}}
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..200
S. Alex Bradt, Jennifer Elder, Pamela E. Harris, Gordon Rojas Kirby, Eva Reutercrona, Yuxuan (Susan) Wang, and Juliet Whidden, Unit interval parking functions and the r-Fubini numbers, arXiv:2401.06937 [math.CO], 2024. See page 10.
Jennifer Elder, Pamela E. Harris, Jan Kretschmann, and J. Carlos Martínez Mori, Boolean intervals in the weak order of S_n, arXiv:2306.14734 [math.CO], 2023.
Laura Gellert and Raman Sanyal, On degree sequences of undirected, directed, and bidirected graphs, arXiv preprint arXiv:1512.08448 [math.CO], 2015.
Hannah Golab, Pattern avoidance in Cayley permutations, Master's Thesis, Northern Arizona Univ. (2024). See p. 36.
Harri Hakula, Helmut Harbrecht, Vesa Kaarnioja, Frances Y. Kuo, and Ian H. Sloan, Uncertainty quantification for random domains using periodic random variables, arXiv:2210.17329 [math.NA], 2022.
Dixy Msapato, Counting the number of tau-exceptional sequences over Nakayama algebras, arXiv:2002.12194 [math.RT], 2020.
Robert A. Proctor, Let's Expand Rota's Twelvefold Way For Counting Partitions!, arXiv:math/0606404 [math.CO], 2006-2007.
Gus Wiseman, Sequences counting and ranking compositions by the patterns they match or avoid.
Index entries for related partition-counting sequences

Cf. A000085, A000670, A002530, A002605, A005442, A052585, A052611.
Cf. A090932, A102726, A106356, A232464, A333755, A335455, A335456.
Column k=2 of A276921.
Cubefree numbers are A004709.
(1,1)-avoiding patterns are A000142.
(1,1,1)-avoiding compositions are A232432.
(1,1,1)-matching patterns are A335508.
(1,1,1)-avoiding permutations of prime indices are A335511.
(1,1,1)-avoiding compositions are ranked by A335513.
(1,1,1,1)-avoiding patterns are A189886.

[n le 2 select 1 else (n-1)*Self(n-1) + Binomial(n-1,2)*Self(n-2): n in [1..31]]; // G. C. Greubel, Jan 31 2023

a:= n-> n! *(Matrix([[1,1], [1/2,0]])^n)[1,1]:
seq(a(n), n=0..40);  # Alois P. Heinz, Jun 01 2009
a:= gfun:-rectoproc({a(n) = n*a(n-1)+(n*(n-1)/2)*a(n-2),a(0)=1,a(1)=1},a(n),remember):
seq(a(n), n=0..40); # Robert Israel, Nov 01 2015

Table[n!*SeriesCoefficient[-2/(-2+2*x+x^2),{x,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 13 2012 *)
Round@Table[n! ((1+Sqrt[3])^(n+1) - (1-Sqrt[3])^(n+1))/(2^(n+1) Sqrt[3]), {n, 0, 20}] (* Vladimir Reshetnikov, Oct 31 2015 *)

Vec(serlaplace((2/(2-2*x-x^2) + O(x^30)))) \\ Michel Marcus, Nov 02 2015

A002605=BinaryRecurrenceSequence(2,2,0,1)
def A080599(n): return factorial(n)*A002605(n+1)/2^n
[A080599(n) for n in range(41)] # G. C. Greubel, Jan 31 2023

a(n) = n*a(n-1) + (n*(n-1)/2)*a(n-2). - Vladeta Jovovic, Aug 22 2003
E.g.f.: 1/(1-x-x^2/2). - Richard Stanley, May 09 2011
a(n) ~ n!*((1+sqrt(3))/2)^(n+1)/sqrt(3). - Vaclav Kotesovec, Oct 13 2012
a(n) = n!*((1+sqrt(3))^(n+1) - (1-sqrt(3))^(n+1))/(2^(n+1)*sqrt(3)). - Vladimir Reshetnikov, Oct 31 2015
a(n) = A090932(n) * A002530(n+1). - Robert Israel, Nov 01 2015

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.

A344652 Number of permutations of the prime indices of n with no adjacent triples (..., x, y, z, ...) such that x <= y <= z.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 0, 1, 2, 1, 2, 1, 2, 2, 0, 1, 2, 1, 2, 2, 2, 1, 1, 1, 2, 0, 2, 1, 5, 1, 0, 2, 2, 2, 3, 1, 2, 2, 1, 1, 5, 1, 2, 2, 2, 1, 0, 1, 2, 2, 2, 1, 1, 2, 1, 2, 2, 1, 7, 1, 2, 2, 0, 2, 5, 1, 2, 2, 5, 1, 2, 1, 2, 2, 2, 2, 5, 1, 0, 0, 2, 1, 7, 2, 2, 2

Offset: 1

Gus Wiseman, Jun 17 2021

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.

			The permutations for n = 2, 6, 8, 30, 36, 60, 180, 210, 360:
  (1)  (12)  (132)  (1212)  (1213)  (12132)  (1324)  (121213)
       (21)  (213)  (2121)  (1312)  (13212)  (1423)  (121312)
             (231)  (2211)  (1321)  (13221)  (1432)  (121321)
             (312)          (2131)  (21213)  (2143)  (131212)
             (321)          (2311)  (21312)  (2314)  (132121)
                            (3121)  (21321)  (2413)  (132211)
                            (3211)  (22131)  (2431)  (212131)
                                    (23121)  (3142)  (213121)
                                    (23211)  (3214)  (213211)
                                    (31212)  (3241)  (221311)
                                    (32121)  (3412)  (231211)
                                    (32211)  (3421)  (312121)
                                             (4132)  (321211)
                                             (4213)
                                             (4231)
                                             (4312)
                                             (4321)

All permutations of prime indices are counted by A008480.
The case of permutations is A049774.
Avoiding (3,2,1) also gives A344606.
The wiggly case is A345164.
A001250 counts wiggly permutations.
A025047 counts wiggly compositions (ascend: A025048, descend: A025049).
A056239 adds up prime indices, row sums of A112798.
A325534 counts separable partitions, ranked by A335433.
A325535 counts inseparable partitions, ranked by A335448.
A335452 counts anti-run permutations of prime indices.
A345170 counts partitions with a wiggly permutation, ranked by A345172.
A345192 counts non-wiggly compositions, ranked by A345168.
Counting compositions by patterns:
- A102726 avoiding (1,2,3).
- A128761 avoiding (1,2,3) adjacent.
- A335514 matching (1,2,3).
- A344614 avoiding (1,2,3) and (3,2,1) adjacent.
- A344615 weakly avoiding (1,2,3) adjacent.
Cf. A001222, A003242, A056986, A316524, A333213, A335511, A344604, A344653, A344654, A345167, A345173.

Table[Length[Select[Permutations[Flatten[ ConstantArray@@@FactorInteger[n]]],!MatchQ[#,{_,x_,y_,z_,_}/;x<=y<=z]&]],{n,100}]

A232432 Number of compositions of n avoiding the pattern 111.

Original entry on oeis.org

1, 1, 2, 3, 7, 11, 21, 34, 59, 114, 178, 284, 522, 823, 1352, 2133, 3739, 5807, 9063, 14074, 23639, 36006, 56914, 87296, 131142, 214933, 324644, 487659, 739291, 1108457, 1724673, 2558386, 3879335, 5772348, 8471344, 12413666, 19109304, 27886339, 40816496

Offset: 0

Alois P. Heinz, Nov 23 2013

nonn

Number of compositions of n into parts with multiplicity <= 2.

			a(4) = 7: [4], [3,1], [2,2], [1,3], [2,1,1], [1,2,1], [1,1,2].
a(5) = 11: [5], [4,1], [3,2], [2,3], [1,4], [3,1,1], [2,2,1], [1,3,1], [2,1,2], [1,2,2], [1,1,3].
a(6) = 21: [6], [4,2], [3,3], [5,1], [2,4], [1,5], [2,1,3], [1,2,3], [1,1,4], [4,1,1], [3,2,1], [2,3,1], [1,4,1], [3,1,2], [1,3,2], [1,2,2,1], [2,1,1,2], [1,2,1,2], [1,1,2,2], [2,2,1,1], [2,1,2,1].

Alois P. Heinz, 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.

Cf. A000726 (partitions avoiding 111), A032020 (pattern 11), A128695 (adjacent pattern 111).
Column k=2 of A243081.
The case of partitions is ranked by A004709.
The version for patterns is A080599.
(1,1,1,1)-avoiding partitions are counted by A232464.
The (1,1,1)-matching version is A335455.
Patterns matched by compositions are counted by A335456.
The version for prime indices is A335511.
(1,1,1)-avoiding compositions are ranked by A335513.
Cf. A011782, A006918, A102726, A106356, A238279, A333755.

b:= proc(n, i, p) option remember; `if`(n=0, p!, `if`(i<1, 0,
      add(b(n-i*j, i-1, p+j)/j!, j=0..min(n/i, 2))))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..50);

f[list_]:=Apply[And,Table[Count[list,i]<3,{i,1,Max[list]}]];
g[list_]:=Length[list]!/Apply[Times,Table[Count[list,i]!,{i,1,Max[list]}]];
a[n_] := If[n == 0, 1, Total[Map[g, Select[IntegerPartitions[n], f]]]];
Table[a[n], {n, 0, 35}] (* Geoffrey Critzer, Nov 25 2013, updated by Jean-François Alcover, Nov 20 2023 *)

A335510 Number of (1,1,1)-matching permutations of the prime indices of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 4, 0, 4, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 10, 0, 0, 0, 0, 0, 0, 0, 5, 1, 0, 0, 0, 0, 0, 0

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.

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.

Patterns matching this pattern are counted by A335508.
These compositions are counted by A335455.
The (1,1)-matching version is A335487.
The complement A335511 is the avoiding version.
These permutations are ranked by A335512.
Permutations of prime indices are counted by A008480.
Patterns are counted by A000670 and ranked by A333217.
Anti-run permutations of prime indices are counted by A335452.
Cf. A056239, A056986, A112798, A238279, A281188, A333221, A333755, A335456, A335460, A335462, A335463.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[Select[Permutations[primeMS[n]],MatchQ[#,{_,x_,_,x_,_,x_,_}]&]],{n,0,100}]

If n is cubefree, a(n) = 0; otherwise a(n) = A008480(n).

A335513 Numbers k such that the k-th composition in standard order (A066099) avoids the pattern (1,1,1).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 14, 16, 17, 18, 19, 20, 21, 22, 24, 25, 26, 28, 32, 33, 34, 35, 36, 37, 38, 40, 41, 43, 44, 45, 46, 48, 49, 50, 52, 53, 54, 56, 58, 64, 65, 66, 67, 68, 69, 70, 72, 73, 74, 75, 76, 77, 78, 80, 81, 82, 83, 84, 88, 89

Offset: 1

Gus Wiseman, Jun 18 2020

nonn

These are compositions with no 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:
   0: ()         17: (4,1)      37: (3,2,1)
   1: (1)        18: (3,2)      38: (3,1,2)
   2: (2)        19: (3,1,1)    40: (2,4)
   3: (1,1)      20: (2,3)      41: (2,3,1)
   4: (3)        21: (2,2,1)    43: (2,2,1,1)
   5: (2,1)      22: (2,1,2)    44: (2,1,3)
   6: (1,2)      24: (1,4)      45: (2,1,2,1)
   8: (4)        25: (1,3,1)    46: (2,1,1,2)
   9: (3,1)      26: (1,2,2)    48: (1,5)
  10: (2,2)      28: (1,1,3)    49: (1,4,1)
  11: (2,1,1)    32: (6)        50: (1,3,2)
  12: (1,3)      33: (5,1)      52: (1,2,3)
  13: (1,2,1)    34: (4,2)      53: (1,2,2,1)
  14: (1,1,2)    35: (4,1,1)    54: (1,2,1,2)
  16: (5)        36: (3,3)      56: (1,1,4)

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

These compositions are counted by A232432 (by sum).
The (1,1)-avoiding version is A233564.
The complement A335512 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.
Patterns avoiding (1,1,1) are counted by A080599 (by length).
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.
Permutations of prime indices avoiding (1,1,1) are counted by A335511.
Cf. A034691, A056986, A108917, 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_,_}]&]

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A080599 Expansion of e.g.f.: 2/(2-2*x-x^2).

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

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A344652 Number of permutations of the prime indices of n with no adjacent triples (..., x, y, z, ...) such that x <= y <= z.

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A232432 Number of compositions of n avoiding the pattern 111.

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A335510 Number of (1,1,1)-matching permutations of the prime indices of n.

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A335513 Numbers k such that the k-th composition in standard order (A066099) avoids the pattern (1,1,1).

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