A335488 -id:A335488 - cp's OEIS frontend

A261982 Number of compositions of n with some part repeated.

0, 0, 1, 1, 5, 11, 21, 51, 109, 229, 455, 959, 1947, 3963, 7999, 16033, 32333, 64919, 130221, 260967, 522733, 1045825, 2093855, 4189547, 8382315, 16768455, 33543127, 67093261, 134193413, 268404995, 536829045, 1073686083, 2147408773, 4294869253, 8589803783

Offset: 0

Alois P. Heinz, Sep 07 2015

nonn

Also compositions matching the pattern (1,1). - Gus Wiseman, Jun 23 2020

			a(2) = 1: 11.
a(3) = 1: 111.
a(4) = 5: 22, 211, 121, 112, 1111.

Alois P. Heinz, Table of n, a(n) for n = 0..3322
Wikipedia, Permutation pattern
Gus Wiseman, Sequences counting and ranking compositions by the patterns they match or avoid.

Row sums of A261981 and of A262191.
Cf. A262047.
The version for patterns is A019472.
The (1,1)-avoiding version is A032020.
The case of partitions is A047967.
(1,1,1)-matching compositions are counted by A335455.
Patterns matched by compositions are counted by A335456.
(1,1)-matching compositions are ranked by A335488.
Cf. A011782, A056986, A106356, A181796, A238279, A269134, A333755.

b:= proc(n, k) option remember; `if`(k<0 or n<0, 0,
      `if`(k=0, `if`(n=0, 1, 0), b(n-k, k) +k*b(n-k, k-1)))
    end:
a:= n-> ceil(2^(n-1))-add(b(n, k), k=0..floor((sqrt(8*n+1)-1)/2)):
seq(a(n), n=0..40);

b[n_, k_] := b[n, k] = If[k<0 || n<0, 0, If[k==0, If[n==0, 1, 0], b[n-k, k] + k*b[n-k, k-1]]]; a[n_] := Ceiling[2^(n-1)]-Sum[b[n, k], {k, 0, Floor[ (Sqrt[8n+1]-1)/2]}]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Feb 08 2017, translated from Maple *)
Table[Length[Join@@Permutations/@Select[IntegerPartitions[n],Length[#]>Length[Split[#]]&]],{n,0,10}] (* Gus Wiseman, Jun 24 2020 *)

a(n) = A011782(n) - A032020(n).

G.f.: (1 - x) / (1 - 2*x) - Sum_{k>=0} k! * x^(k*(k + 1)/2) / Product_{j=1..k} (1 - x^j). - Ilya Gutkovskiy, Jan 30 2020

A019472 Weak preference orderings of n alternatives, i.e., orderings that have indifference between at least two alternatives.

Original entry on oeis.org

0, 0, 1, 7, 51, 421, 3963, 42253, 505515, 6724381, 98618763, 1582715773, 27612565995, 520631327581, 10554164679243, 228975516609853, 5294731892093355, 130015079601039901, 3379132289551117323, 92679942218919579133, 2675254894236207563115, 81073734056332364441821

Offset: 0

Robert Ware (bware(AT)wam.umd.edu)

From Gus Wiseman, Jun 24 2020: (Start)
Equivalently, a(n) is number of (1,1)-matching sequences of length n that cover an initial interval of positive integers. For example, the a(2) = 1 and a(3) = 7 sequences are:
  (1,1)  (1,1,1)
         (1,1,2)
         (1,2,1)
         (1,2,2)
         (2,1,1)
         (2,1,2)
         (2,2,1)
Missing from this list are:
  (1,2)  (1,2,3)
  (2,1)  (1,3,2)
         (2,1,3)
         (2,3,1)
         (3,1,2)
         (3,2,1)
(End)

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

Cf. A000670, A052875.
(1,1)-avoiding patterns are counted by A000142.
(1,2)-matching patterns are counted by A056823.
(1,1)-matching compositions are counted by A261982.
(1,1)-matching compositions are ranked by A335488.
Patterns matched by patterns are counted by A335517.
Cf. A056986, A333217, A335454, A335456, A335515.

a[n_] := Sum[(-1)^(j-i)*Binomial[j, i]*i^n, {i, 0, n-1}, {j, 0, n-1}]; Table[a[n], {n, 0, 21}] (* Jean-François Alcover, Feb 26 2016, after Peter Luschny *)

def A019472(n):
    return add(add((-1)^(j-i)*binomial(j, i)*i^n for i in range(n)) for j in range(n))
[A019472(n) for n in range(21)] # Peter Luschny, Jul 22 2014

a(n) = A000670(n) - n!. - corrected by Eugene McDonnell, May 12 2000

a(n) = Sum_{j=0..n-1} Sum_{i=0..n-1} (-1)^(j-i)*C(j, i)*i^n. - Peter Luschny, Jul 22 2014

A321773 Number of compositions of n into parts with distinct multiplicities and with exactly three parts.

Original entry on oeis.org

1, 3, 6, 4, 9, 9, 10, 12, 15, 13, 18, 18, 19, 21, 24, 22, 27, 27, 28, 30, 33, 31, 36, 36, 37, 39, 42, 40, 45, 45, 46, 48, 51, 49, 54, 54, 55, 57, 60, 58, 63, 63, 64, 66, 69, 67, 72, 72, 73, 75, 78, 76, 81, 81, 82, 84, 87, 85, 90, 90, 91, 93, 96, 94, 99, 99

Offset: 3

Alois P. Heinz, Nov 18 2018

nonn

			From _Gus Wiseman_, Nov 11 2020: (Start)
Also the number of 3-part non-strict compositions of n. For example, the a(3) = 1 through a(11) = 15 triples are:
  111   112   113   114   115   116   117   118   119
        121   122   141   133   161   144   181   155
        211   131   222   151   224   171   226   191
              212   411   223   233   225   244   227
              221         232   242   252   262   272
              311         313   323   333   334   335
                          322   332   414   343   344
                          331   422   441   424   353
                          511   611   522   433   434
                                      711   442   443
                                            622   515
                                            811   533
                                                  551
                                                  722
                                                  911
(End)

Alois P. Heinz, Table of n, a(n) for n = 3..1000

Column k=3 of A242887.
A235451 counts 3-part compositions with distinct run-lengths
A001399(n-6) counts 3-part compositions in the complement.
A014311 intersected with A335488 ranks these compositions.
A140106 is the unordered case, with Heinz numbers A285508.
A261982 counts non-strict compositions of any length.
A001523 counts unimodal compositions, with complement A115981.
A007318 and A097805 count compositions by length.
A032020 counts strict compositions.
A047967 counts non-strict partitions, with Heinz numbers A013929.
A242771 counts triples that are not strictly increasing.
Cf. A000212, A000217, A001840, A128422, A156040, A332834, A337461, A337484, A337603, A337604.

Table[Length[Join@@Permutations/@Select[IntegerPartitions[n,{3}],!UnsameQ@@#&]],{n,0,100}] (* Gus Wiseman, Nov 11 2020 *)

Conjectures from Colin Barker, Dec 11 2018: (Start)
G.f.: x^3*(1 + 3*x + 5*x^2) / ((1 - x)^2*(1 + x)*(1 + x + x^2)).
a(n) = a(n-2) + a(n-3) - a(n-5) for n>7. (End)
Conjecture: a(n) = (3*n-k)/2 where k value has a cycle of 6 starting from n=3 of (7,6,3,10,3,6). - Bill McEachen, Aug 12 2025

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_,_}]&]

A335487 Number of (1,1)-matching permutations of the prime indices of n.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 3, 0, 0, 0, 1, 0, 3, 0, 3, 0, 0, 0, 4, 1, 0, 1, 3, 0, 0, 0, 1, 0, 0, 0, 6, 0, 0, 0, 4, 0, 0, 0, 3, 3, 0, 0, 5, 1, 3, 0, 3, 0, 4, 0, 4, 0, 0, 0, 12, 0, 0, 3, 1, 0, 0, 0, 3, 0, 0, 0, 10, 0, 0, 3, 3, 0, 0, 0, 5, 1, 0, 0, 12, 0, 0

Offset: 1

Gus Wiseman, Jun 14 2020

nonn

Depends 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 = 4, 12, 24, 48, 36, 72, 60:
  (11)  (112)  (1112)  (11112)  (1122)  (11122)  (1123)
        (121)  (1121)  (11121)  (1212)  (11212)  (1132)
        (211)  (1211)  (11211)  (1221)  (11221)  (1213)
               (2111)  (12111)  (2112)  (12112)  (1231)
                       (21111)  (2121)  (12121)  (1312)
                                (2211)  (12211)  (1321)
                                        (21112)  (2113)
                                        (21121)  (2131)
                                        (21211)  (2311)
                                        (22111)  (3112)
                                                 (3121)
                                                 (3211)

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

Positions of zeros are A005117 (squarefree numbers).
The case where the match must be contiguous is A333175.
The avoiding version is A335489.
The (1,1,1)-matching case is A335510.
Patterns are counted by A000670.
Permutations of prime indices are counted by A008480.
(1,1)-matching patterns are counted by A019472.
(1,1)-matching compositions are counted by A261982.
STC-numbers of permutations of prime indices are A333221.
Patterns matched by standard compositions are counted by A335454.
Dimensions of downsets of standard compositions are A335465.
(1,1)-matching compositions are ranked by A335488.
Cf. A000961, A056239, A056986, A112798, A181796, A335451, A335452, A335460, A335462.

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

a(n) = 0 if n is squarefree, otherwise a(n) = A008480(n).

a(n) = A008480(n) - A281188(n) for n != 4.

cp's OEIS Frontend

A261982 Number of compositions of n with some part repeated.

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A019472 Weak preference orderings of n alternatives, i.e., orderings that have indifference between at least two alternatives.

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A321773 Number of compositions of n into parts with distinct multiplicities and with exactly three parts.

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

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A335487 Number of (1,1)-matching permutations of the prime indices of n.

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