A002051 -id:A002051 - cp's OEIS frontend

A056823 Number of compositions minus number of partitions: A011782(n) - A000041(n).

0, 0, 0, 1, 3, 9, 21, 49, 106, 226, 470, 968, 1971, 3995, 8057, 16208, 32537, 65239, 130687, 261654, 523661, 1047784, 2096150, 4193049, 8387033, 16775258, 33551996, 67105854, 134214010, 268430891, 536865308, 1073734982, 2147475299, 4294957153, 8589922282

Offset: 0

Alford Arnold, Aug 29 2000

Previous name was: Counts members of A056808 by number of factors.
A056808 relates to least prime signatures (cf. A025487)
a(n) is also the number of compositions of n that are not partitions of n. - Omar E. Pol, Jan 31 2009, Oct 14 2013
a(n) is the number of compositions of n into positive parts containing pattern [1,2]. - Bob Selcoe, Jul 08 2014

			A011782 begins     1 1 2 4 8 16 32 64 128 256 ...;
A000041 begins     1 1 2 3 5  7 11 15  22  30 ...;
so sequence begins 0 0 0 1 3  9 21 49 106 226 ... .
For n = 3 the factorizations are 8=2*2*2, 12=2*2*3, 18=2*3*3 and 30=2*3*5.
a(5) = 9: {[1,1,1,2], [1,1,2,1], [1,1,3], [1,2,1,1], [1,2,2], [1,3,1], [1,4], [2,1,2], [2,3]}. - _Bob Selcoe_, Jul 08 2014

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

Cf. A025487, A056808, A117989.
The version for patterns is A002051.
(1,2)-avoiding compositions are just partitions A000041.
The (1,1)-matching version is A261982.
The version for prime indices is A335447.
(1,2)-matching compositions are ranked by A335485.
Patterns matched by compositions are counted by A335456.
Cf. A011782, A056986, A106356, A144300, A238279, A269134, A333755.

a:= n-> ceil(2^(n-1))-combinat[numbpart](n):
seq(a(n), n=0..37);  # Alois P. Heinz, Jan 30 2020

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],!GreaterEqual@@#&]],{n,0,10}] (* Gus Wiseman, Jun 24 2020 *)
a[n_] := If[n == 0, 0, 2^(n-1) - PartitionsP[n]];
a /@ Range[0, 37] (* Jean-François Alcover, May 23 2021 *)

a(n) = A011782(n) - A000041(n).
a(n) = 2*a(n-1) + A117989(n-1). - Bob Selcoe, Apr 11 2014
G.f.: (1 - x) / (1 - 2*x) - Product_{k>=1} 1 / (1 - x^k). - Ilya Gutkovskiy, Jan 30 2020

More terms from James Sellers, Aug 31 2000

New name from Joerg Arndt, Sep 02 2013

A335485 Numbers k such that the k-th composition in standard order (A066099) is not weakly decreasing.

Original entry on oeis.org

6, 12, 13, 14, 20, 22, 24, 25, 26, 27, 28, 29, 30, 38, 40, 41, 44, 45, 46, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 70, 72, 76, 77, 78, 80, 81, 82, 83, 84, 86, 88, 89, 90, 91, 92, 93, 94, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106

Offset: 1

Gus Wiseman, Jun 18 2020

nonn

Also compositions matching the pattern (1,2).

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.

			The sequence of terms together with the corresponding compositions begins:
   6: (1,2)
  12: (1,3)
  13: (1,2,1)
  14: (1,1,2)
  20: (2,3)
  22: (2,1,2)
  24: (1,4)
  25: (1,3,1)
  26: (1,2,2)
  27: (1,2,1,1)
  28: (1,1,3)
  29: (1,1,2,1)
  30: (1,1,1,2)
  38: (3,1,2)
  40: (2,4)

Keiichi Shigechi, Noncommutative crossing partitions, arXiv:2211.10958 [math.CO], 2022.
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 A114994 is the avoiding version.
The (2,1)-matching version is A335486.
Patterns matching this pattern are counted by A002051 (by length).
Permutations of prime indices matching this pattern are counted by A335447.
These compositions are counted by A056823 (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, A335456, A335458.

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

A335486 Numbers k such that the k-th composition in standard order (A066099) is not weakly increasing.

Original entry on oeis.org

5, 9, 11, 13, 17, 18, 19, 21, 22, 23, 25, 27, 29, 33, 34, 35, 37, 38, 39, 41, 43, 44, 45, 46, 47, 49, 50, 51, 53, 54, 55, 57, 59, 61, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 81, 82, 83, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 97, 98, 99

Offset: 1

Gus Wiseman, Jun 18 2020

nonn

Also compositions matching the pattern (2,1).

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.

			The sequence of terms together with the corresponding compositions begins:
   5: (2,1)
   9: (3,1)
  11: (2,1,1)
  13: (1,2,1)
  17: (4,1)
  18: (3,2)
  19: (3,1,1)
  21: (2,2,1)
  22: (2,1,2)
  23: (2,1,1,1)
  25: (1,3,1)
  27: (1,2,1,1)
  29: (1,1,2,1)
  33: (5,1)
  34: (4,2)
  35: (4,1,1)

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 A225620 is the avoiding version.
The (1,2)-matching version is A335485.
Patterns matching this pattern are counted by A002051 (by length).
Permutations of prime indices matching this pattern are counted by A008480(n) - 1.
These compositions are counted by A056823 (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, A334968, A335456, A335458.

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

A241168 Triangle read by rows: T(n,k) (1 <= k <= n) = Steffensen's bracket function [n,n-k].

Original entry on oeis.org

1, 1, 2, 1, 5, 6, 1, 9, 25, 26, 1, 14, 67, 149, 150, 1, 20, 145, 525, 1081, 1082, 1, 27, 275, 1450, 4651, 9365, 9366, 1, 35, 476, 3430, 15421, 47229, 94585, 94586, 1, 44, 770, 7266, 43281, 180894, 545707, 1091669, 1091670, 1, 54, 1182, 14154, 107751, 581280, 2359225, 7087005, 14174521, 14174522

Offset: 1

N. J. A. Sloane, Apr 22 2014

Steffensen's bracket function [n,k] = Sum_{s=k..n-1} Stirling2(n,s+1)*s!/k!.

The numbers are used in numerical integration.

			Triangle begins:
1,
1, 2,
1, 5, 6,
1, 9, 25, 26,
1, 14, 67, 149, 150,
1, 20, 145, 525, 1081, 1082,
1, 27, 275, 1450, 4651, 9365, 9366,
1, 35, 476, 3430, 15421, 47229, 94585, 94586,
1, 44, 770, 7266, 43281, 180894, 545707, 1091669, 1091670,
...

J. F. Steffensen, On a class of polynomials and their application to actuarial problems, Skandinavisk Aktuarietidskrift, 11 (1928), 75-97.

J. F. Steffensen, On a class of polynomials and their application to actuarial problems, Skandinavisk Aktuarietidskrift, Vol. 11, pp. 75-97, 1928.

Diagonals include A000096, A000629, A002050, A002051, A241169, A241170.

with(combinat);
T:=proc(n,k) add(stirling2(n,s+1)*s!/k!,s=k..n-1); end;
for n from 1 to 12 do lprint([seq(T(n,n-k),k=1..n)]); od:

A335447 Number of (1,2)-matching permutations of the prime indices of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 1, 1, 0, 0, 2, 0, 2, 1, 1, 0, 3, 0, 1, 0, 2, 0, 5, 0, 0, 1, 1, 1, 5, 0, 1, 1, 3, 0, 5, 0, 2, 2, 1, 0, 4, 0, 2, 1, 2, 0, 3, 1, 3, 1, 1, 0, 11, 0, 1, 2, 0, 1, 5, 0, 2, 1, 5, 0, 9, 0, 1, 2, 2, 1, 5, 0, 4, 0, 1, 0, 11, 1, 1

Offset: 1

Gus Wiseman, Jun 14 2020

nonn

Depends only on sorted prime signature (A118914).
Also the number of (2,1)-matching permutations of the prime indices of n.
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 = 6, 12, 24, 48, 30, 72, 60:
  (12)  (112)  (1112)  (11112)  (123)  (11122)  (1123)
        (121)  (1121)  (11121)  (132)  (11212)  (1132)
               (1211)  (11211)  (213)  (11221)  (1213)
                       (12111)  (231)  (12112)  (1231)
                                (312)  (12121)  (1312)
                                       (12211)  (1321)
                                       (21112)  (2113)
                                       (21121)  (2131)
                                       (21211)  (2311)
                                                (3112)
                                                (3121)

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

The avoiding version is A000012.
Patterns are counted by A000670.
Positions of zeros are A000961.
(1,2)-matching patterns are counted by A002051.
Permutations of prime indices are counted by A008480.
(1,2)-matching compositions are counted by A056823.
STC-numbers of permutations of prime indices are A333221.
Patterns matched by standard compositions are counted by A335454.
(1,2,1) or (2,1,2)-matching permutations of prime indices are A335460.
(1,2,1) and (2,1,2)-matching permutations of prime indices are A335462.
Dimensions of downsets of standard compositions are A335465.
(1,2)-matching compositions are ranked by A335485.
Cf. A056239, A056986, A112798, A181796, A333175, A335451, A335452, A335463, A333175.

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

a(n) = A008480(n) - 1.

A294032 Triangle read by rows, T(n, k) = Pochhammer(3, k)*Stirling2(3 + n, 3 + k) for n >= 0 and 0 <= k <= n.

Original entry on oeis.org

1, 6, 3, 25, 30, 12, 90, 195, 180, 60, 301, 1050, 1680, 1260, 360, 966, 5103, 12600, 15960, 10080, 2520, 3025, 23310, 83412, 158760, 166320, 90720, 20160, 9330, 102315, 510300, 1369620, 2116800, 1890000, 907200, 181440

Offset: 0

Peter Luschny, Oct 22 2017

			Triangle starts:
[0]    1
[1]    6,      3
[2]   25,     30,     12
[3]   90,    195,    180,      60
[4]  301,   1050,   1680,    1260,     360
[5]  966,   5103,  12600,   15960,   10080,    2520
[6] 3025,  23310,  83412,  158760,  166320,   90720,  20160
[7] 9330, 102315, 510300, 1369620, 2116800, 1890000, 907200, 181440

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

T(n, 0) = A000392(n+3), T(n, n) = A001710(n+2).
Row sums A002051(n+3), alternating row sums A000225(n+1).
Cf. A028246 (m=1), A053440 (m=2), this seq. (m=3), A293617 (hub).

A294032 := (n, k) -> pochhammer(3, k)*Stirling2(n + 3, k + 3):
seq(seq(A294032(n, k), k=0..n), n=0..7);
T := (n, k) -> A293617(3, n, k): seq(seq(T(n, k), k=0..n), n=0..7);

Table[Pochhammer[3, k] StirlingS2[3 + n, 3 + k], {n, 0, 7}, {k, 0, n}] // Flatten (* Michael De Vlieger, Oct 22 2017 *)

for(n=0,10, for(k=0,n, print1((k+2)!*stirling(n+3,k+3,2)/2, ", "))) \\ G. C. Greubel, Nov 19 2017

E.g.f.: (1/2)*exp(x)*(2*y + 9*exp(2*x) + y^2+1-11*exp(3*x)*y + 15*y^2*exp(2*x) - 7*y^2*exp(x) - 13*y^2*exp(3*x) + 4*exp(4*x)*y^2 - 8*exp(x) + 24*y*exp(2*x) - 15*y*exp(x))/(1 - y*(exp(x) - 1))^3.

T(n, k) = A293617(3, n, k).

A054255 Triangle T(n,k) (n >= 1, 0<=k<=n) giving number of preferential arrangements of n things beginning with k (transposed, then read by rows).

Original entry on oeis.org

1, 1, 2, 2, 5, 6, 6, 18, 25, 26, 24, 84, 134, 149, 150, 120, 480, 870, 1050, 1081, 1082, 720, 3240, 6600, 8700, 9302, 9365, 9366, 5040, 25200, 57120, 82320, 92526, 94458, 94585, 94586, 40320, 221760, 554400, 871920, 1038744, 1085364, 1091414, 1091669, 1091670

Offset: 1

Eugene McDonnell (Eemcd(AT)aol.com), May 05 2000

Can be generated from Stirling_2 triangle A008277 (cf. A028246, which is intermediate between the two arrays).

			   1;
   1,  2;
   2,  5,   6;
   6, 18,  25,  26;
  24, 84, 134, 149, 150;
  ...

N. J. A. Sloane and Thomas Wieder, The Number of Hierarchical Orderings, Order 21 (2004), 83-89.

Row sums give A000670. First 3 rows are A000629, A002050 = A000629 - 1, 2*A002051 = (A000629 - 2^m) (m >= 0).

Cf. A090665 (triangle with rows reversed).

More terms from James Sellers, May 05 2000

A375408 Numbers k such that the k-th composition in standard order is not weakly increasing or weakly decreasing.

Original entry on oeis.org

13, 22, 25, 27, 29, 38, 41, 44, 45, 46, 49, 50, 51, 53, 54, 55, 57, 59, 61, 70, 76, 77, 78, 81, 82, 83, 86, 88, 89, 90, 91, 92, 93, 94, 97, 98, 99, 101, 102, 103, 105, 107, 108, 109, 110, 111, 113, 114, 115, 117, 118, 119, 121, 123, 125, 134, 140, 141, 142

Offset: 1

Gus Wiseman, Sep 18 2024

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.

			The terms and corresponding compositions begin:
  13: (1,2,1)
  22: (2,1,2)
  25: (1,3,1)
  27: (1,2,1,1)
  29: (1,1,2,1)
  38: (3,1,2)
  41: (2,3,1)
  44: (2,1,3)
  45: (2,1,2,1)
  46: (2,1,1,2)
  49: (1,4,1)
  50: (1,3,2)
  51: (1,3,1,1)
  53: (1,2,2,1)
  54: (1,2,1,2)
  55: (1,2,1,1,1)
  57: (1,1,3,1)
  59: (1,1,2,1,1)

Eric Weisstein's World of Mathematics, Unimodal Sequence.

The version for run-lengths of compositions is A332833.
Compositions of this type are counted by A332834, complement maybe A329398.
A001523 counts unimodal compositions, ranks too dense.
A011782 counts compositions.
A114994 ranks weakly decreasing compositions, complement A335485.
A115981 counts non-unimodal compositions, ranked by A335373.
A225620 ranks weakly increasing compositions, complement A335486.
A238130, A238279, A333755 count compositions by number of runs.
A332835 counts compositions with weakly incr. or weakly decr. run-lengths.
All of the following pertain to compositions in standard order:
- Length is A000120.
- Sum is A029837(n+1).
- Parts are listed by A066099.
- Number of adjacent equal pairs is A124762, unequal A333382.
- Number of max runs: A124765, A124766, A124767, A124768, A124769, A333381.
- Ranks of strict compositions are A233564.
- Ranks of constant compositions are A272919.
- Anti-runs are ranked by A333489, counted by A003242.
- Run-length transform is A333627, sum A070939.
Cf. A001511, A002051, A065120, A329744, A332745, A332836, A332870, A333218.

stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,100],!LessEqual@@stc[#]&&!GreaterEqual@@stc[#]&]

Intersection of A335485 and A335486.

cp's OEIS Frontend

A056823 Number of compositions minus number of partitions: A011782(n) - A000041(n).

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A335485 Numbers k such that the k-th composition in standard order (A066099) is not weakly decreasing.

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A335486 Numbers k such that the k-th composition in standard order (A066099) is not weakly increasing.

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A241168 Triangle read by rows: T(n,k) (1 <= k <= n) = Steffensen's bracket function [n,n-k].

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A335447 Number of (1,2)-matching permutations of the prime indices of n.

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A294032 Triangle read by rows, T(n, k) = Pochhammer(3, k)*Stirling2(3 + n, 3 + k) for n >= 0 and 0 <= k <= n.

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A054255 Triangle T(n,k) (n >= 1, 0<=k<=n) giving number of preferential arrangements of n things beginning with k (transposed, then read by rows).

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A375408 Numbers k such that the k-th composition in standard order is not weakly increasing or weakly decreasing.

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