A333382 -id:A333382 - cp's OEIS frontend

A238279 Triangle read by rows: T(n,k) is the number of compositions of n into nonzero parts with k parts directly followed by a different part, n>=0, 0<=k<=A004523(n-1).

1, 1, 2, 2, 2, 3, 4, 1, 2, 10, 4, 4, 12, 14, 2, 2, 22, 29, 10, 1, 4, 26, 56, 36, 6, 3, 34, 100, 86, 31, 2, 4, 44, 148, 200, 99, 16, 1, 2, 54, 230, 374, 278, 78, 8, 6, 58, 322, 680, 654, 274, 52, 2, 2, 74, 446, 1122, 1390, 814, 225, 22, 1, 4, 88, 573, 1796, 2714, 2058, 813, 136, 10, 4, 88, 778, 2694, 4927

Offset: 0

Joerg Arndt and Alois P. Heinz, Feb 22 2014

Same as A238130, with zeros omitted.
Last elements in rows are 1, 1, 2, 2, 1, 4, 2, 1, 6, 2, 1, 8, ... with g.f. -(x^6+x^4-2*x^2-x-1)/(x^6-2*x^3+1).
For n > 0, also the number of compositions of n with k + 1 runs. - Gus Wiseman, Apr 10 2020

			Triangle starts:
  00:  1;
  01:  1;
  02:  2;
  03:  2,   2;
  04:  3,   4,   1;
  05:  2,  10,   4;
  06:  4,  12,  14,    2;
  07:  2,  22,  29,   10,    1;
  08:  4,  26,  56,   36,    6;
  09:  3,  34, 100,   86,   31,    2;
  10:  4,  44, 148,  200,   99,   16,    1;
  11:  2,  54, 230,  374,  278,   78,    8;
  12:  6,  58, 322,  680,  654,  274,   52,    2;
  13:  2,  74, 446, 1122, 1390,  814,  225,   22,   1;
  14:  4,  88, 573, 1796, 2714, 2058,  813,  136,  10;
  15:  4,  88, 778, 2694, 4927, 4752, 2444,  618,  77,  2;
  16:  5, 110, 953, 3954, 8531, 9930, 6563, 2278, 415, 28, 1;
  ...
Row n=5 is 2, 10, 4 because in the 16 compositions of 5
  ##:  [composition]  no. of changes
  01:  [ 1 1 1 1 1 ]   0
  02:  [ 1 1 1 2 ]   1
  03:  [ 1 1 2 1 ]   2
  04:  [ 1 1 3 ]   1
  05:  [ 1 2 1 1 ]   2
  06:  [ 1 2 2 ]   1
  07:  [ 1 3 1 ]   2
  08:  [ 1 4 ]   1
  09:  [ 2 1 1 1 ]   1
  10:  [ 2 1 2 ]   2
  11:  [ 2 2 1 ]   1
  12:  [ 2 3 ]   1
  13:  [ 3 1 1 ]   1
  14:  [ 3 2 ]   1
  15:  [ 4 1 ]   1
  16:  [ 5 ]   0
there are 2 with no changes, 10 with one change, and 4 with two changes.

Joerg Arndt and Alois P. Heinz, Rows n = 0..180, flattened

Columns k=0-10 give: A000005 (for n>0), 2*A002133, A244714, A244715, A244716, A244717, A244718, A244719, A244720, A244721, A244722.
Row lengths are A004523.
Row sums are A011782.
The version counting adjacent equal parts is A106356.
The version for ascents/descents is A238343.
The version for weak ascents/descents is A333213.
The k-th composition in standard-order has A124762(k) adjacent equal parts, A124767(k) maximal runs, A333382(k) adjacent unequal parts, and A333381(k) maximal anti-runs.
Cf. A064113, A333214, A333216.

b:= proc(n, v) option remember; `if`(n=0, 1, expand(
      add(b(n-i, i)*`if`(v=0 or v=i, 1, x), i=1..n)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0)):
seq(T(n), n=0..14);

b[n_, v_] := b[n, v] = If[n == 0, 1, Expand[Sum[b[n-i, i]*If[v == 0 || v == i, 1, x], {i, 1, n}]]]; T[n_] := Function[{p}, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, 0]]; Table[T[n], {n, 0, 14}] // Flatten (* Jean-François Alcover, Feb 11 2015, after Maple *)
Table[If[n==0,1,Length[Select[Join@@Permutations/@IntegerPartitions[n],Length[Split[#]]==k+1&]]],{n,0,12},{k,0,If[n==0,0,Floor[2*(n-1)/3]]}] (* Gus Wiseman, Apr 10 2020 *)

T_xy(max_row) = {my(N=max_row+1, x='x+O('x^N),h=(1+ sum(i=1,N,(x^i-y*x^i)/(1+y*x^i-x^i)))/(1-sum(i=1,N, y*x^i/(1+y*x^i-x^i)))); for(n=0,N-1, print(Vecrev(polcoeff(h,n))))}
T_xy(16) \\ John Tyler Rascoe, Jul 10 2024

G.f.: A(x,y) = ( 1 + Sum_{i>0} ((x^i)*(1 - y)/(1 + y*x^i - x^i)) )/( 1 - Sum_{i>0} ((y*x^i)/(1 + y*x^i - x^i)) ). - John Tyler Rascoe, Jul 10 2024

A333489 Numbers k such that the k-th composition in standard order is an anti-run (no adjacent equal parts).

Original entry on oeis.org

0, 1, 2, 4, 5, 6, 8, 9, 12, 13, 16, 17, 18, 20, 22, 24, 25, 32, 33, 34, 37, 38, 40, 41, 44, 45, 48, 49, 50, 52, 54, 64, 65, 66, 68, 69, 70, 72, 76, 77, 80, 81, 82, 88, 89, 96, 97, 98, 101, 102, 104, 105, 108, 109, 128, 129, 130, 132, 133, 134, 137, 140, 141

Offset: 1

Gus Wiseman, Mar 28 2020

nonn

A composition of n is a finite sequence of positive integers summing to n. The k-th composition in standard order (row k of 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.

			The sequence together with the corresponding compositions begins:
    0: ()          33: (5,1)         70: (4,1,2)
    1: (1)         34: (4,2)         72: (3,4)
    2: (2)         37: (3,2,1)       76: (3,1,3)
    4: (3)         38: (3,1,2)       77: (3,1,2,1)
    5: (2,1)       40: (2,4)         80: (2,5)
    6: (1,2)       41: (2,3,1)       81: (2,4,1)
    8: (4)         44: (2,1,3)       82: (2,3,2)
    9: (3,1)       45: (2,1,2,1)     88: (2,1,4)
   12: (1,3)       48: (1,5)         89: (2,1,3,1)
   13: (1,2,1)     49: (1,4,1)       96: (1,6)
   16: (5)         50: (1,3,2)       97: (1,5,1)
   17: (4,1)       52: (1,2,3)       98: (1,4,2)
   18: (3,2)       54: (1,2,1,2)    101: (1,3,2,1)
   20: (2,3)       64: (7)          102: (1,3,1,2)
   22: (2,1,2)     65: (6,1)        104: (1,2,4)
   24: (1,4)       66: (5,2)        105: (1,2,3,1)
   25: (1,3,1)     68: (4,3)        108: (1,2,1,3)
   32: (6)         69: (4,2,1)      109: (1,2,1,2,1)

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000

Anti-runs summing to n are counted by A003242(n).
A triangle counting maximal anti-runs of compositions is A106356.
A triangle counting maximal runs of compositions is A238279 or A238130.
Partitions whose first differences are an anti-run are A238424.
All of the following pertain to compositions in standard order (A066099):
- Adjacent equal pairs are counted by A124762.
- Weakly decreasing runs are counted by A124765.
- Weakly increasing runs are counted by A124766.
- Equal runs are counted by A124767.
- Strictly increasing runs are counted by A124768.
- Strictly decreasing runs are counted by A124769.
- Strict compositions are ranked by A233564.
- Constant compositions are ranked by A272919.
- Normal compositions are ranked by A333217.
- Anti-runs are counted by A333381.
- Adjacent unequal pairs are counted by A333382.
Cf. A000120, A029931, A048793, A066099, A070939, A228351.

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

A106356 Triangle T(n,k) 0<=k

Original entry on oeis.org

1, 1, 1, 3, 0, 1, 4, 3, 0, 1, 7, 6, 2, 0, 1, 14, 7, 8, 2, 0, 1, 23, 20, 10, 8, 2, 0, 1, 39, 42, 22, 13, 9, 2, 0, 1, 71, 72, 58, 28, 14, 10, 2, 0, 1, 124, 141, 112, 72, 33, 16, 11, 2, 0, 1, 214, 280, 219, 150, 92, 36, 18, 12, 2, 0, 1, 378, 516, 466, 311, 189, 112, 40, 20, 13, 2, 0, 1

Offset: 1

Christian G. Bower, Apr 29 2005

For n > 0, also the number of compositions of n with k + 1 maximal anti-runs (sequences without adjacent equal terms). - Gus Wiseman, Mar 23 2020

			T(4,1) = 3 because the compositions of 4 with 1 adjacent equal part are 1+1+2, 2+1+1, 2+2.
Triangle begins:
   1;
   1,  1;
   3,  0,  1;
   4,  3,  0, 1;
   7,  6,  2, 0, 1;
  14,  7,  8, 2, 0, 1;
  23, 20, 10, 8, 2, 0, 1;
  ...
From _Gus Wiseman_, Mar 23 2020 (Start)
Row n = 6 counts the following compositions (empty column shown by dot):
  (6)     (33)    (222)    (11112)  .  (111111)
  (15)    (114)   (1113)   (21111)
  (24)    (411)   (1122)
  (42)    (1131)  (2211)
  (51)    (1221)  (3111)
  (123)   (1311)  (11121)
  (132)   (2112)  (11211)
  (141)           (12111)
  (213)
  (231)
  (312)
  (321)
  (1212)
  (2121)
(End)

Alois P. Heinz, Rows n = 1..141, flattened
A. Knopfmacher and H. Prodinger, On Carlitz compositions, European Journal of Combinatorics, Vol. 19, 1998, pp. 579-589.

Row sums: 2^(n-1)=A000079(n-1). Columns 0-4: A003242, A106357-A106360.
The version counting adjacent unequal parts is A238279.
The k-th composition in standard-order has A124762(k) adjacent equal parts and A333382(k) adjacent unequal parts.
The k-th composition in standard-order has A124767(k) maximal runs and A333381(k) maximal anti-runs.
The version for ascents/descents is A238343.
The version for weak ascents/descents is A333213.
Cf. A064113, A066099, A233564, A333214, A333216.

b:= proc(n, h, t) option remember;
      if n=0 then `if`(t=0, 1, 0)
    elif t<0 then 0
    else add(b(n-j, j, `if`(j=h, t-1, t)), j=1..n)
      fi
    end:
T:= (n, k)-> b(n, -1, k):
seq(seq(T(n, k), k=0..n-1), n=1..15); # Alois P. Heinz, Oct 23 2011

b[n_, h_, t_] := b[n, h, t] = If[n == 0, If[t == 0, 1, 0], If[t<0, 0, Sum[b[n-j, j, If [j == h, t-1, t]], {j, 1, n}]]]; T[n_, k_] := b[n, -1, k]; Table[Table[T[n, k], {k, 0, n-1}], {n, 1, 15}] // Flatten (* Jean-François Alcover, Feb 20 2015, after Alois P. Heinz *)
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],n==0||Length[Split[#,#1!=#2&]]==k+1&]],{n,0,12},{k,0,n}] (* Gus Wiseman, Mar 23 2020 *)

A124767 Number of level runs for compositions in standard order.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 2, 1, 1, 2, 1, 2, 2, 3, 2, 1, 1, 2, 2, 2, 2, 2, 3, 2, 2, 3, 2, 3, 2, 3, 2, 1, 1, 2, 2, 2, 1, 3, 3, 2, 2, 3, 1, 2, 3, 4, 3, 2, 2, 3, 3, 3, 3, 3, 4, 3, 2, 3, 2, 3, 2, 3, 2, 1, 1, 2, 2, 2, 2, 3, 3, 2, 2, 2, 2, 3, 3, 4, 3, 2, 2, 3, 3, 3, 2, 2, 3, 2, 3, 4, 3, 4, 3, 4, 3, 2, 2, 3, 3, 3, 2, 4, 4, 3, 3

Offset: 0

Franklin T. Adams-Watters, Nov 06 2006

The standard order of compositions is given by A066099.

For n > 0, a(n) is one more than the number of adjacent unequal terms in the n-th composition in standard order. Also the number of runs in the same composition. - Gus Wiseman, Apr 08 2020

			Composition number 11 is 2,1,1; the level runs are 2; 1,1; so a(11) = 2.
The table starts:
  0
  1
  1 1
  1 2 2 1
  1 2 1 2 2 3 2 1
  1 2 2 2 2 2 3 2 2 3 2 3 2 3 2 1
  1 2 2 2 1 3 3 2 2 3 1 2 3 4 3 2 2 3 3 3 3 3 4 3 2 3 2 3 2 3 2 1
The 1234567th composition in standard order is (3,2,1,2,2,1,2,5,1,1,1) with runs ((3),(2),(1),(2,2),(1),(2),(5),(1,1,1)), so a(1234567) = 8. - _Gus Wiseman_, Apr 08 2020

Row-lengths are A011782.
Compositions counted by number of runs are A238279 or A333755.
All of the following pertain to compositions in standard order (A066099):
- Length is A000120.
- Sum is A070939.
- Weakly decreasing compositions are A114994.
- Adjacent equal pairs are counted by A124762.
- Weakly decreasing runs are counted by A124765.
- Weakly increasing runs are counted by A124766.
- Equal runs are counted by A124767 (this sequence).
- Weakly increasing compositions are A225620.
- Strict compositions A233564.
- Constant compositions are A272919.
- Anti-runs are counted by A333381.
- Adjacent unequal pairs are counted by A333382.
- Anti-run compositions are A333489.
- Runs-resistance is A333628.
- Run-lengths are A333769 (triangle).
Cf. A029931, A048793, A066099, A106356, A228351, A318928, A329744, A333217, A333219, A333627.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Table[Length[Split[stc[n]]],{n,0,100}] (* Gus Wiseman, Apr 17 2020 *)

a(0) = 0, a(n) = 1 + Sum_{1<=i=1 0.

For n > 0, a(n) = A333382(n) + 1. - Gus Wiseman, Apr 08 2020

A333755 Triangle read by rows where T(n,k) is the number of compositions of n with k runs, n >= 0, 0 <= k <= n.

Original entry on oeis.org

1, 0, 1, 0, 2, 0, 0, 2, 2, 0, 0, 3, 4, 1, 0, 0, 2, 10, 4, 0, 0, 0, 4, 12, 14, 2, 0, 0, 0, 2, 22, 29, 10, 1, 0, 0, 0, 4, 26, 56, 36, 6, 0, 0, 0, 0, 3, 34, 100, 86, 31, 2, 0, 0, 0, 0, 4, 44, 148, 200, 99, 16, 1, 0, 0, 0, 0, 2, 54, 230, 374, 278, 78, 8, 0, 0, 0, 0

Offset: 0

Gus Wiseman, Apr 10 2020

Except for a(1) = 0, the data is identical to A238130 shifted right once. However, in A238130, each row after the first ends with a zero, while here each row after the first starts with a zero.

			Triangle begins:
   1
   0   1
   0   2   0
   0   2   2   0
   0   3   4   1   0
   0   2  10   4   0   0
   0   4  12  14   2   0   0
   0   2  22  29  10   1   0   0
   0   4  26  56  36   6   0   0   0
   0   3  34 100  86  31   2   0   0   0
   0   4  44 148 200  99  16   1   0   0   0
   0   2  54 230 374 278  78   8   0   0   0   0
Row n = 6 counts the following compositions (empty column indicated by dot):
  .  (6)       (15)     (123)    (1212)
     (33)      (24)     (132)    (2121)
     (222)     (42)     (141)
     (111111)  (51)     (213)
               (114)    (231)
               (411)    (312)
               (1113)   (321)
               (1122)   (1131)
               (2211)   (1221)
               (3111)   (1311)
               (11112)  (2112)
               (21111)  (11121)
                        (11211)
                        (12111)

Removing all zeros gives A238279.
The version for anti-runs is A106356.
The k-th composition in standard-order has A124767(k) runs.
The version counting descents is A238343.
The version counting weak ascents is A333213.
Cf. A066099, A124762, A238130, A272919, A333381, A333382, A333489.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],Length[Split[#]]==k&]],{n,0,10},{k,0,n}]

A333381 Number of maximal anti-runs of the n-th composition in standard order.

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 1, 3, 1, 1, 2, 2, 1, 1, 2, 4, 1, 1, 1, 2, 1, 2, 1, 3, 1, 1, 2, 2, 2, 2, 3, 5, 1, 1, 1, 2, 2, 1, 1, 3, 1, 1, 3, 3, 1, 1, 2, 4, 1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 3, 3, 3, 3, 4, 6, 1, 1, 1, 2, 1, 1, 1, 3, 1, 2, 2, 2, 1, 1, 2, 4, 1, 1, 1, 2, 2, 3, 2

Offset: 0

Gus Wiseman, Mar 24 2020

nonn

Anti-runs are sequences without any adjacent equal terms.
A composition of n is a finite sequence of positive integers summing to n. The k-th composition in standard order (row k of 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.
For n > 0, also one plus the number of adjacent equal pairs in the n-th composition in standard order.

			The 46th composition in standard order is (2,1,1,2), with maximal anti-runs ((2,1),(1,2)), so a(46) = 2.

Wikipedia, Longest increasing subsequence

Anti-runs summing to n are counted by A003242(n).
A triangle counting maximal anti-runs of compositions is A106356.
A triangle counting maximal runs of compositions is A238279.
Partitions whose first differences are an anti-run are A238424.
All of the following pertain to compositions in standard order (A066099):
- Adjacent equal pairs are counted by A124762.
- Weakly decreasing runs are counted by A124765.
- Weakly increasing runs are counted by A124766.
- Equal runs are counted by A124767.
- Strictly increasing runs are counted by A124768.
- Strictly decreasing runs are counted by A124769.
- Strict compositions are ranked by A233564.
- Constant compositions are ranked by A272919.
- Normal compositions are ranked by A333217.
- Adjacent unequal pairs are counted by A333382.
- Anti-runs are ranked by A333489.
Cf. A000120, A029931, A048793, A059893, A070939, A114994, A225620, A228351.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Table[Length[Split[stc[n],UnsameQ]],{n,0,100}]

For n > 0, a(n) = A124762(n) + 1.

A005649 Expansion of e.g.f. (2 - e^x)^(-2).

Original entry on oeis.org

1, 2, 8, 44, 308, 2612, 25988, 296564, 3816548, 54667412, 862440068, 14857100084, 277474957988, 5584100659412, 120462266974148, 2772968936479604, 67843210855558628, 1757952715142990612, 48093560991292628228, 1385244691781856307124

Offset: 0

N. J. A. Sloane, Simon Plouffe

Exponential self-convolution of numbers of preferential arrangements.
Number of compatible bipartitional relations on a set of cardinality n. - Ralf Stephan, Apr 27 2003
Stirling transform of A000142, shifted left one place: 1, 2, 6, 24, 120, 720, ... - Philippe Deléham, May 17 2005; corrected by Ilya Gutkovskiy, Jul 25 2018
With an extra 1 at the beginning, coefficients of the formal (divergent) series expansion at infinity of Sum_{k>=0} 1/binomial(x,k) = 1+1/x+2/x^2+8/x^3+... Also Sum_{k>=0} k!/x^k Product_{i=1..k-1} 1/(1-i/x) yields a generating function in 1/x. - Roland Bacher, Nov 21 2000
Stirling-Bernoulli transform of A001057: 1, -1, 2, -2, 3, -3, 4, ... - Philippe Deléham, May 27 2015
a(n) is the total number of open sets summed over all chain topologies that can be placed on an n-set.  A chain topology is a topology whose open sets can be totally ordered by inclusion. - Geoffrey Critzer, Apr 06 2017
From Gus Wiseman, Jun 10 2020: (Start)
Also the number of length n + 1 sequences covering an initial interval of positive integers with no adjacent equal parts (anti-runs). For example, the a(0) = 1 through a(2) = 8 anti-runs are:
  (1)  (1,2)  (1,2,1)
       (2,1)  (1,2,3)
              (1,3,2)
              (2,1,2)
              (2,1,3)
              (2,3,1)
              (3,1,2)
              (3,2,1)
Also the number of ordered set partitions of {1,...,n + 1} with no two successive vertices in the same block. For example, the a(0) = 1 through a(2) = 8 ordered set partitions are:
  {{1}}  {{1},{2}}   {{1,3},{2}}
         {{2},{1}}   {{2},{1,3}}
                    {{1},{2},{3}}
                    {{1},{3},{2}}
                    {{2},{1},{3}}
                    {{2},{3},{1}}
                    {{3},{1},{2}}
                    {{3},{2},{1}}
(End)
From Manfred Boergens, Feb 24 2025: (Start)
a(n+1) is the n-th row sum in A380977.
Number of surjections f with domain [n+1] and f(n+1)!=f(j) for jNumber of (n+1)-tuples containing all elements of a set, with a unique last element.
Consider an urn with balls of pairwise different colors. a(n) is the number of (n+1)-sequences of draws with replacement completing the covering of all colors with the last draw, the number of colors running from 1 to n+1.
(End)

			a(2)=8 gives the number of 3-tuples containing all elements of a set [n] with n<=3 and a unique last element: 112, 221, 123, 213, 132, 312, 231, 321. - _Manfred Boergens_, Feb 24 2025

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 294.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 0..423 (first 101 terms from T. D. Noe)
José A. Adell, Beáta Bényi, Venkat Murali, and Sithembele Nkonkobe, Generalized Barred Preferential Arrangements, Transactions on Combinatorics (2022).
Connor Ahlbach, Jeremy Usatine and Nicholas Pippenger, Barred Preferential Arrangements, Electron. J. Combin., Volume 20, Issue 2 (2013), #P55.
D. Foata, and C. Krattenthaler, Graphical Major Indices, II, Séminaire Lotharingien de Combinatoire, B34k, 16 pp., 1995.
D. Foata and D. Zeilberger, The Graphical Major Index, arXiv:math/9406220 [math.CO], 1994.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 154
Augustine O. Munagi, Two Applications of the Bijection on Fibonacci Set Partitions, Fibonacci Quart. 55 (2017), no. 5, 144-148.

Cf. A000670.
2*A083410(n)=a(n), if n>0.
Pairwise sums of A052841 and also of A089677.
Anti-run compositions are counted by A003242.
A triangle counting maximal anti-runs of compositions is A106356.
Anti-runs of standard compositions are counted by A333381.
Adjacent unequal pairs in standard compositions are counted by A333382.
Cf. A000110, A008277, A055932, A124762, A238279, A238424, A317081.
Cf. A380977.

b:= proc(n, m) option remember;
     `if`(n=0, (m+1)!, m*b(n-1, m)+b(n-1, m+1))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..23);  # Alois P. Heinz, Aug 03 2021

f[n_] := Sum[(i + j)^n/2^(2 + i + j), {i, 0, Infinity}, {j, 0, Infinity}]; Array[f, 20, 0] (* Vladimir Reshetnikov, Dec 31 2008 *)
a[n_] := (-1)^n (PolyLog[-n-1, 2] - PolyLog[-n, 2])/4; Array[f, 20, 0] (* Vladimir Reshetnikov, Jan 23 2011 *)
Range[0, 19]! CoefficientList[Series[(2 - Exp@ x)^-2, {x, 0, 19}], x] (* Robert G. Wilson v, Jan 23 2011 *)
nn = 19; Range[0, nn]! CoefficientList[Series[1 + D[u^2 (Exp[z] - 1)/(1 - u (Exp[z] - 1)), u] /. u -> 1, {z, 0, nn}], z] (* Geoffrey Critzer, Apr 06 2017 *)
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],FreeQ[Differences[#],0]&]],{n,0,6}] (* Gus Wiseman, Jun 10 2020 *)
With[{nn=20},CoefficientList[Series[1/(2-E^x)^2,{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Oct 02 2021 *)
Table[Sum[(m+1)! StirlingS2[n,m],{m,0,n}],{n,0,19}] (* Manfred Boergens, Feb 24 2025 *)

t(n):=sum(stirling2(n,k)*k!,k,0,n);
makelist(sum(binomial(n,k)*t(k)*t(n-k),k,0,n),n,0,20);
/* Emanuele Munarini, Oct 02 2012 */

a(n)=if(n<0,0,n!*polcoeff(subst(1/(1-y)^2,y,exp(x+x*O(x^n))-1),n))

a(n)=polcoeff(sum(m=0, n,(2*m)!/m!*x^m/prod(k=1, m,1+(m+k)*x+x*O(x^n))), n)
for(n=0, 20, print1(a(n), ", ")) \\ Paul D. Hanna, Jan 03 2013

E.g.f.: 1/(2-exp(x))^2.
a(n) = (A000670(n) + A000670(n+1)) / 2. - Philippe Deléham, May 16 2005
a(n) = D^n(1/(1-x)^2) evaluated at x = 0, where D is the operator (1+x)*d/dx. Cf. A000670 and A052841. - Peter Bala, Nov 25 2011
E.g.f.: 1/(2-exp(x))^2 = 1/(G(0) + 4), G(k) = 1-4/((2^k)-x*(4^k)/((2^k)*x-(2*k+2)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Dec 15 2011
O.g.f.: Sum_{n>=0} (2*n)!/n! * x^n / Product_{k=1..n} (1 + (n+k)*x). - Paul D. Hanna, Jan 03 2013
G.f.: (G(0) - 1)/(x-1) where G(k) = 1 - (k+1)/(1-k*x)/(1-x/(x-1/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jan 15 2013
G.f.: 1/G(0) where G(k) = 1 - x*(k+2)/( 1 - 2*x*(k+1)/G(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Mar 23 2013
a(n) = Sum_{k = 0..n} A163626(n,k) * A001057(k+1). - Philippe Deléham, May 27 2015
a(n) ~ n! * n / (4 * (log(2))^(n+2)). - Vaclav Kotesovec, Jul 01 2018
a(n) = Sum_{k=0..n} Stirling2(n,k)*(k + 1)!. - Ilya Gutkovskiy, Jul 25 2018
From Seiichi Manyama, Nov 19 2023: (Start)
a(0) = 1; a(n) = Sum_{k=1..n} (k/n + 1) * binomial(n,k) * a(n-k).
a(0) = 1; a(n) = 2*a(n-1) - 2*Sum_{k=1..n-1} (-1)^k * binomial(n-1,k) * a(n-k). (End)

A374629 Irregular triangle listing the leaders of maximal weakly increasing runs in the n-th composition in standard order.

Original entry on oeis.org

1, 2, 1, 3, 2, 1, 1, 1, 4, 3, 1, 2, 2, 1, 1, 1, 1, 1, 1, 5, 4, 1, 3, 2, 3, 1, 2, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 6, 5, 1, 4, 2, 4, 1, 3, 3, 2, 1, 3, 1, 3, 1, 2, 2, 1, 2, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 0

Gus Wiseman, Jul 20 2024

The leaders of maximal weakly increasing runs in a sequence are obtained by splitting it into maximal weakly increasing subsequences and taking the first term of each.

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 58654th composition in standard order is (1,1,3,2,4,1,1,1,2), with maximal weakly increasing runs ((1,1,3),(2,4),(1,1,1,2)), so row 58654 is (1,2,1).
The nonnegative integers, corresponding compositions, and leaders of maximal weakly increasing runs begin:
    0:      () -> ()      15: (1,1,1,1) -> (1)
    1:     (1) -> (1)     16:       (5) -> (5)
    2:     (2) -> (2)     17:     (4,1) -> (4,1)
    3:   (1,1) -> (1)     18:     (3,2) -> (3,2)
    4:     (3) -> (3)     19:   (3,1,1) -> (3,1)
    5:   (2,1) -> (2,1)   20:     (2,3) -> (2)
    6:   (1,2) -> (1)     21:   (2,2,1) -> (2,1)
    7: (1,1,1) -> (1)     22:   (2,1,2) -> (2,1)
    8:     (4) -> (4)     23: (2,1,1,1) -> (2,1)
    9:   (3,1) -> (3,1)   24:     (1,4) -> (1)
   10:   (2,2) -> (2)     25:   (1,3,1) -> (1,1)
   11: (2,1,1) -> (2,1)   26:   (1,2,2) -> (1)
   12:   (1,3) -> (1)     27: (1,2,1,1) -> (1,1)
   13: (1,2,1) -> (1,1)   28:   (1,1,3) -> (1)
   14: (1,1,2) -> (1)     29: (1,1,2,1) -> (1,1)

Gus Wiseman, Sequences counting and ranking compositions by their leaders (for six types of runs).

Row-leaders are A065120.
Row-lengths are A124766.
Row-sums are A374630.
Positions of constant rows are A374633, counted by A374631.
Positions of strict rows are A374768, counted by A374632.
For other types of runs we have A374251, A374515, A374683, A374740, A374757.
Positions of non-weakly decreasing rows are A375137.
A011782 counts compositions.
A238130, A238279, A333755 count compositions by number of runs.
A335456 counts patterns matched by compositions.
All of the following pertain to compositions in standard order:
- Length is A000120.
- Sum is A029837(n+1).
- Leader is A065120.
- Parts are listed by A066099, reverse A228351.
- Number of adjacent equal pairs is A124762, unequal A333382.
- Number of max runs: A124765, A124766, A124767, A124768, A124769, A333381.
- Ranks of anti-run compositions are A333489, counted by A003242.
- Run-length transform is A333627, length A124767, sum A070939.
- Run-compression transform is A373948, sum A373953, excess A373954.
- Ranks of contiguous compositions are A374249, counted by A274174.
- Ranks of non-contiguous compositions are A374253, counted by A335548.
Cf. A046660, A106356, A188920, A189076, A238343, A272919, A333213, A373949, A374634, A374635, A374637, A374701, A375123.

stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Table[First/@Split[stc[n],LessEqual],{n,0,100}]

A124762 Number of levels for compositions in standard order.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 1, 1, 0, 0, 1, 3, 0, 0, 0, 1, 0, 1, 0, 2, 0, 0, 1, 1, 1, 1, 2, 4, 0, 0, 0, 1, 1, 0, 0, 2, 0, 0, 2, 2, 0, 0, 1, 3, 0, 0, 0, 1, 0, 1, 0, 2, 1, 1, 2, 2, 2, 2, 3, 5, 0, 0, 0, 1, 0, 0, 0, 2, 0, 1, 1, 1, 0, 0, 1, 3, 0, 0, 0, 1, 1, 2, 1, 3, 0, 0, 1, 1, 1, 1, 2, 4, 0, 0, 0, 1, 1, 0, 0, 2, 0

Offset: 0

Franklin T. Adams-Watters, Nov 06 2006

The standard order of compositions is given by A066099.

A composition of n is a finite sequence of positive integers summing to n. The k-th composition in standard order (row k of 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. This sequence gives the number of adjacent equal terms in the n-th composition in standard order. Alternatively, a(n) is one fewer than the number of maximal anti-runs in the same composition, where anti-runs are sequences without any adjacent equal terms. For example, the 1234567th composition in standard order is (3,2,1,2,2,1,2,5,1,1,1) with anti-runs ((3,2,1,2),(2,1,2,5,1),(1),(1)), so a(1234567) = 4 - 1 = 3. - Gus Wiseman, Apr 08 2020

			Composition number 11 is 2,1,1; 2>1=1, so a(11) = 1.
The table starts:
  0
  0
  0 1
  0 0 0 2
  0 0 1 1 0 0 1 3
  0 0 0 1 0 1 0 2 0 0 1 1 1 1 2 4
  0 0 0 1 1 0 0 2 0 0 2 2 0 0 1 3 0 0 0 1 0 1 0 2 1 1 2 2 2 2 3 5

Cf. A066099, A124760, A124761, A124763, A124764, A011782 (row lengths), A059570 (row sums).
Anti-runs summing to n are counted by A003242(n).
A triangle counting maximal anti-runs of compositions is A106356.
A triangle counting maximal runs of compositions is A238279.
Partitions whose first differences are an anti-run are A238424.
All of the following pertain to compositions in standard order (A066099):
- Weakly decreasing runs are counted by A124765.
- Weakly increasing runs are counted by A124766.
- Equal runs are counted by A124767.
- Strictly increasing runs are counted by A124768.
- Strictly decreasing runs are counted by A124769.
- Strict compositions are A233564.
- Constant compositions are A272919.
- Normal compositions are A333217.
- Adjacent unequal pairs are counted by A333382.
- Anti-runs are A333489.
Cf. A000120, A029931, A048793, A059893, A070939, A114994, A225620, A228351.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Table[Length[Select[Partition[stc[n],2,1],SameQ@@#&]],{n,0,100}] (* Gus Wiseman, Apr 08 2020 *)

For a composition b(1),...,b(k), a(n) = Sum_{1<=i=1

For n > 0, a(n) = A333381(n) - 1. - Gus Wiseman, Apr 08 2020

A374249 Numbers k such that the k-th composition in standard order has its equal parts contiguous.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, 19, 20, 21, 23, 24, 26, 28, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 47, 48, 50, 52, 56, 58, 60, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 78, 79, 80, 81, 83, 84, 85

Offset: 1

Gus Wiseman, Jul 13 2024

nonn

These are compositions avoiding the patterns (1,2,1) and (2,1,2).

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 together with their standard compositions begin:
   0: ()
   1: (1)
   2: (2)
   3: (1,1)
   4: (3)
   5: (2,1)
   6: (1,2)
   7: (1,1,1)
   8: (4)
   9: (3,1)
  10: (2,2)
  11: (2,1,1)
  12: (1,3)
  14: (1,1,2)
  15: (1,1,1,1)
  16: (5)
See A374253 for the complement: 13, 22, 25, 27, 29, ...

Gus Wiseman, Statistics, classes, and transformations of standard compositions

The strict (also anti-run) case is A233564, counted by A032020.
Compositions of this type are counted by A274174.
Permutations of prime indices of this type are counted by A333175.
The complement is A374253 (anti-run A374254), counted by A335548.
A003242 counts anti-run compositions, ranks A333489.
A011782 counts compositions.
A066099 lists compositions in standard order.
A124767 counts runs in standard compositions, anti-runs A333381.
A333755 counts compositions by number of runs.
A335454 counts patterns matched by standard compositions.
A335462 counts (1,2,1)- and (2,1,2)-matching permutations of prime indices.
- A335470 counts (1,2,1)-matching compositions, ranks A335466.
- A335471 counts (1,2,1)-avoiding compositions, ranks A335467.
- A335472 counts (2,1,2)-matching compositions, ranks A335468.
- A335473 counts (2,1,2)-avoiding compositions, ranks A335469.
Cf. A106356, A124762, A238130, A238279, A261982, A272919, A333382, A335450, A335460, A335524, A335525.

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

Equals A335467 /\ A335469.

Showing 1-10 of 69 results. Next

cp's OEIS Frontend

A238279 Triangle read by rows: T(n,k) is the number of compositions of n into nonzero parts with k parts directly followed by a different part, n>=0, 0<=k<=A004523(n-1).

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A333489 Numbers k such that the k-th composition in standard order is an anti-run (no adjacent equal parts).

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A106356 Triangle T(n,k) 0<=k

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A124767 Number of level runs for compositions in standard order.

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A333755 Triangle read by rows where T(n,k) is the number of compositions of n with k runs, n >= 0, 0 <= k <= n.

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A333381 Number of maximal anti-runs of the n-th composition in standard order.

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A005649 Expansion of e.g.f. (2 - e^x)^(-2).

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