A238424 -id:A238424 - cp's OEIS frontend

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

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)

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

A333382 Number of adjacent unequal parts in the n-th composition in standard-order.

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, Mar 24 2020

nonn

For n > 0, a(n) is one fewer than the number of maximal runs of the n-th composition in standard-order.

			The 46th composition in standard order is (2,1,1,2), with maximal runs ((2),(1,1),(2)), so a(46) = 3 - 1 = 2.

Wikipedia, Longest increasing subsequence

Indices of first appearances (not counting 0) are A113835.
Partitions whose 0-appended first differences are a run are A007862.
Partitions whose first differences are a run are A049988.
A triangle counting maximal anti-runs of compositions is A106356.
A triangle counting maximal runs of compositions is A238279.
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 ranked by A333489.
- Anti-runs are counted by A333381.
Cf. A000005, A000120, A003242, A029931, A048793, A059893, A070939, A114994, A225620, A228351, A238424.

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

For n > 0, a(n) = A124767(n) - 1.

A374515 Irregular triangle read by rows where row n lists the leaders of anti-runs in the n-th composition in standard order.

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, Jul 31 2024

Anti-runs summing to n are counted by A003242(n).
The leaders of anti-runs in a sequence are obtained by splitting it into maximal consecutive anti-runs (sequences with no adjacent equal terms) 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 maximal anti-runs of the 1234567th composition in standard order are ((3,2,1,2),(2,1,2,5,1),(1),(1)), so row 1234567 is (3,2,1,1).
The nonnegative integers, corresponding compositions, and leaders of anti-runs begin:
    0:      () -> ()        15: (1,1,1,1) -> (1,1,1,1)
    1:     (1) -> (1)       16:       (5) -> (5)
    2:     (2) -> (2)       17:     (4,1) -> (4)
    3:   (1,1) -> (1,1)     18:     (3,2) -> (3)
    4:     (3) -> (3)       19:   (3,1,1) -> (3,1)
    5:   (2,1) -> (2)       20:     (2,3) -> (2)
    6:   (1,2) -> (1)       21:   (2,2,1) -> (2,2)
    7: (1,1,1) -> (1,1,1)   22:   (2,1,2) -> (2)
    8:     (4) -> (4)       23: (2,1,1,1) -> (2,1,1)
    9:   (3,1) -> (3)       24:     (1,4) -> (1)
   10:   (2,2) -> (2,2)     25:   (1,3,1) -> (1)
   11: (2,1,1) -> (2,1)     26:   (1,2,2) -> (1,2)
   12:   (1,3) -> (1)       27: (1,2,1,1) -> (1,1)
   13: (1,2,1) -> (1)       28:   (1,1,3) -> (1,1)
   14: (1,1,2) -> (1,1)     29: (1,1,2,1) -> (1,1)

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

Row-leaders of nonempty rows are A065120.
Row-lengths are A333381.
Row-sums are A374516.
Positions of identical rows are A374519 (counted by A374517).
Positions of distinct (strict) rows are A374638 (counted by A374518).
A106356 counts compositions by number of maximal anti-runs.
A238279 counts compositions by number of maximal runs
A238424 counts partitions whose first differences are an anti-run.
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.
- Anti-runs are ranked by A333489, counted by A003242.
- Run-length transform is A333627, sum A070939.
- Run-compression is A373948 or A374251, sum A373953, excess A373954.
- Ranks of contiguous compositions are A374249, counted by A274174.
Six types of maximal runs:
- Count: A124766, A124765, A124768, A124769, A333381, A124767.
- Leaders: A374629, A374740, A374683, A374757, A374515, A374251.
- Rank: A375123, A375124, A375125, A375126, A375127, A373948.
Cf. A029931, A059893, A228351, A233564, A272919, A333213, A373949.

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

A374518 Number of integer compositions of n whose leaders of anti-runs are distinct.

Original entry on oeis.org

1, 1, 1, 3, 5, 9, 17, 32, 58, 112, 201, 371, 694, 1276, 2342, 4330, 7958, 14613, 26866, 49303, 90369, 165646, 303342, 555056, 1015069, 1855230

Offset: 0

Gus Wiseman, Aug 01 2024

nonn

The leaders of anti-runs in a sequence are obtained by splitting it into maximal consecutive anti-runs (sequences with no adjacent equal terms) and taking the first term of each.

			The a(0) = 1 through a(6) = 17 compositions:
  ()  (1)  (2)  (3)   (4)    (5)    (6)
                (12)  (13)   (14)   (15)
                (21)  (31)   (23)   (24)
                      (121)  (32)   (42)
                      (211)  (41)   (51)
                             (122)  (123)
                             (131)  (132)
                             (212)  (141)
                             (311)  (213)
                                    (231)
                                    (312)
                                    (321)
                                    (411)
                                    (1212)
                                    (1221)
                                    (2112)
                                    (2121)

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

These compositions have ranks A374638.
The complement is counted by A374678.
For partitions instead of compositions we have A375133.
Other types of runs (instead of anti-):
- For leaders of identical runs we have A274174, ranks A374249.
- For leaders of weakly increasing runs we have A374632, ranks A374768.
- For leaders of strictly increasing runs we have A374687, ranks A374698.
- For leaders of weakly decreasing runs we have A374743, ranks A374701.
- For leaders of strictly decreasing runs we have A374761, ranks A374767.
Other types of run-leaders (instead of distinct):
- For identical leaders we have A374517.
- For weakly increasing leaders we have A374681.
- For strictly increasing leaders we have A374679.
- For weakly decreasing leaders we have A374682.
- For strictly decreasing leaders we have A374680.
A003242 counts anti-runs, ranks A333489.
A106356 counts compositions by number of maximal anti-runs.
A238279 counts compositions by number of maximal runs
A238424 counts partitions whose first differences are an anti-run.
Cf. A188920, A233564, A238343, A333213, A333381, A373949, A374515.

Table[Length[Select[Join@@Permutations /@ IntegerPartitions[n],UnsameQ@@First/@Split[#,UnsameQ]&]],{n,0,15}]

A374517 Number of integer compositions of n whose leaders of anti-runs are identical.

Original entry on oeis.org

1, 1, 2, 4, 7, 13, 25, 46, 85, 160, 301, 561, 1056, 1984, 3730, 7037, 13273, 25056, 47382, 89666, 169833, 322038, 611128, 1160660, 2206219, 4196730, 7988731, 15217557, 29005987, 55321015, 105570219, 201569648, 385059094, 735929616, 1407145439, 2691681402

Offset: 0

Gus Wiseman, Aug 01 2024

nonn

The leaders of anti-runs in a sequence are obtained by splitting it into maximal consecutive anti-runs (sequences with no adjacent equal terms) and taking the first term of each.

			The a(0) = 1 through a(5) = 13 compositions:
  ()  (1)  (2)   (3)    (4)     (5)
           (11)  (12)   (13)    (14)
                 (21)   (22)    (23)
                 (111)  (31)    (32)
                        (112)   (41)
                        (121)   (113)
                        (1111)  (131)
                                (212)
                                (221)
                                (1112)
                                (1121)
                                (1211)
                                (11111)

John Tyler Rascoe, Table of n, a(n) for n = 0..100
Gus Wiseman, Sequences counting and ranking compositions by their leaders (for six types of runs).

For partitions instead of compositions we have A034296 or A115029.
These compositions have ranks A374519.
The complement is counted by A374640.
Other types of runs (instead of anti-):
- For leaders of identical runs we have A000005 for n > 0, ranks A272919.
- For leaders of weakly increasing runs we have A374631, ranks A374633.
- For leaders of strictly increasing runs we have A374686, ranks A374685.
- For leaders of weakly decreasing runs we have A374742, ranks A374741.
- For leaders of strictly decreasing runs we have A374760, ranks A374759.
Other types of run-leaders (instead of identical):
- For distinct leaders we have A374518.
- For weakly increasing leaders we have A374681.
- For strictly increasing leaders we have A374679.
- For weakly decreasing leaders we have A374682.
- For strictly decreasing leaders we have A374680.
A003242 counts anti-runs, ranks A333489.
A106356 counts compositions by number of maximal anti-runs.
A238279 counts compositions by number of maximal runs
A238424 counts partitions whose first differences are an anti-run.
A274174 counts contiguous compositions, ranks A374249.
Cf. A188920, A189076, A238343, A333213, A373949, A374515, A374518.

Table[Length[Select[Join@@Permutations /@ IntegerPartitions[n],SameQ@@First/@Split[#,UnsameQ]&]],{n,0,15}]

C_x(N) = {my(g =1/(1 - sum(k=1, N, x^k/(1+x^k))));g}
A_x(i,N) = {my(x='x+O('x^N), f=(x^i)*(C_x(N)*(x^i)+x^i+1)/(1+x^i)^2);f}
B_x(i,j,N) = {my(x='x+O('x^N), f=C_x(N)*x^(i+j)/((1+x^i)*(1+x^j)));f}
D_x(N) = {my(x='x+O('x^N), f=1+sum(i=1,N,-1+sum(j=0,N-i, A_x(i,N)^j)*(1-B_x(i,i,N)+sum(k=1,N-i,B_x(i,k,N)))));Vec(f)}
D_x(30) \\ John Tyler Rascoe, Aug 16 2024

G.f.: 1 + Sum_{i>0} (-1 + Sum_{j>=0} (A(i,x)^j)*(1 + Sum_{k>0, k<>i} (B(i,k,x)))) where A(i,x) = (x^i)*(C(x)*(x^i) + x^i + 1)/(1+x^i)^2, B(i,k,x) = C(x)*x^(i+k)/((1+x^i)*(1+x^k)), and C(x) is the g.f. for A003242. - John Tyler Rascoe, Aug 16 2024

a(26) onwards from John Tyler Rascoe, Aug 16 2024

A374638 Numbers k such that the leaders of anti-runs in the k-th composition in standard order (A066099) are distinct.

Original entry on oeis.org

0, 1, 2, 4, 5, 6, 8, 9, 11, 12, 13, 16, 17, 18, 19, 20, 22, 24, 25, 26, 32, 33, 34, 35, 37, 38, 40, 41, 44, 45, 46, 48, 49, 50, 52, 53, 54, 64, 65, 66, 67, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 81, 82, 83, 88, 89, 91, 92, 93, 96, 97, 98, 100, 101, 102, 104

Offset: 1

Gus Wiseman, Aug 01 2024

nonn

The leaders of anti-runs in a sequence are obtained by splitting it into maximal consecutive anti-runs (sequences with no adjacent equal terms) 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 terms together with corresponding compositions begin:
   0: ()
   1: (1)
   2: (2)
   4: (3)
   5: (2,1)
   6: (1,2)
   8: (4)
   9: (3,1)
  11: (2,1,1)
  12: (1,3)
  13: (1,2,1)
  16: (5)
  17: (4,1)
  18: (3,2)
  19: (3,1,1)
  20: (2,3)
  22: (2,1,2)
  24: (1,4)
  25: (1,3,1)
  26: (1,2,2)

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

Positions of distinct (strict) rows in A374515.
Compositions of this type are counted by A374518.
For identical instead of distinct we have A374519, counted by A374517.
The complement is A374639.
Other types of runs (instead of anti-):
- For identical runs we have A374249, counted by A274174.
- For weakly increasing runs we have A374768, counted by A374632.
- For strictly increasing runs we have A374698, counted by A374687.
- For weakly decreasing runs we have A374701, counted by A374743.
- For strictly decreasing runs we have A374767, counted by A374761.
A065120 gives leaders of standard compositions.
A106356 counts compositions by number of maximal anti-runs.
A238279 counts compositions by number of maximal runs
A238424 counts partitions whose first differences are an anti-run.
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.
- Anti-runs are ranked by A333489, counted by A003242.
- Run-length transform is A333627, sum A070939.
- Run-compression transform is A373948, sum A373953, excess A373954.
Six types of maximal runs:
- Count: A124766, A124765, A124768, A124769, A333381, A124767.
- Leaders: A374629, A374740, A374683, A374757, A374515, A374251.
- Rank: A375123, A375124, A375125, A375126, A375127, A373948.
Cf. A029931, A114994, A228351, A233564, A238343, A272919, A335467, A373949.

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

Showing 1-10 of 30 results. Next

Definition: a partition is separable if there is an ordering of its parts in which no consecutive parts are identical; otherwise the partition is inseparable.

A partition with k parts is inseparable if and only if there is a part whose multiplicity is greater than ceiling(k/2). - Andrew Howroyd, Jan 17 2024

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.

cp's OEIS Frontend

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

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A325535 Number of inseparable partitions of n; see Comments.

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PARI

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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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Maple

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Maxima

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A124762 Number of levels for compositions in standard order.

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A333382 Number of adjacent unequal parts in the n-th composition in standard-order.

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A374515 Irregular triangle read by rows where row n lists the leaders of anti-runs in the n-th composition in standard order.

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