A106356 -id:A106356 - cp's OEIS frontend

A351017 Number of binary words of length n with all distinct run-lengths.

1, 2, 2, 6, 6, 10, 22, 26, 38, 54, 114, 130, 202, 266, 386, 702, 870, 1234, 1702, 2354, 3110, 5502, 6594, 9514, 12586, 17522, 22610, 31206, 48630, 60922, 83734, 111482, 149750, 196086, 261618, 336850, 514810, 631946, 862130, 1116654, 1502982, 1916530, 2555734, 3242546

Offset: 0

Gus Wiseman, Feb 07 2022

nonn

			The a(0) = 1 through a(6) = 22 words:
  {}  0   00   000   0000   00000   000000
      1   11   001   0001   00001   000001
               011   0111   00011   000011
               100   1000   00111   000100
               110   1110   01111   000110
               111   1111   10000   001000
                            11000   001110
                            11100   001111
                            11110   011000
                            11111   011100
                                    011111
                                    100000
                                    100011
                                    100111
                                    110000
                                    110001
                                    110111
                                    111001
                                    111011
                                    111100
                                    111110
                                    111111

Using binary expansions instead of words gives A032020, ranked by A044813.
The version for partitions is A098859.
The complement is counted by twice A261982.
The version for compositions is A329739, for runs A351013.
For runs instead of run-lengths we have A351016, twice A351018.
The version for patterns is A351292, for runs A351200.
A000120 counts binary weight.
A001037 counts binary Lyndon words, necklaces A000031, aperiodic A027375.
A005811 counts runs in binary expansion.
A011782 counts integer compositions.
A242882 counts compositions with distinct multiplicities.
A297770 counts distinct runs in binary expansion.
A325545 counts compositions with distinct differences.
A329767 counts binary words by runs-resistance.
A351014 counts distinct runs in standard compositions.
A351204 counts partitions where every permutation has all distinct runs.
A351290 ranks compositions with all distinct runs.
Cf. A003242, A098504, A106356, A116608, A175413, A233564, A238130 or A238279, A328592, A334028, A350952.

Table[Length[Select[Tuples[{0,1},n],UnsameQ@@Length/@Split[#]&]],{n,0,10}]

from itertools import groupby, product
def adrl(s):
    runlens = [len(list(g)) for k, g in groupby(s)]
    return len(runlens) == len(set(runlens))
def a(n):
    if n == 0: return 1
    return 2*sum(adrl("1"+"".join(w)) for w in product("01", repeat=n-1))
print([a(n) for n in range(20)]) # Michael S. Branicky, Feb 08 2022

a(n>0) = 2 * A032020(n).

a(25)-a(32) from Michael S. Branicky, Feb 08 2022

More terms from David A. Corneth, Feb 08 2022 using data from A032020

A374761 Number of integer compositions of n whose leaders of strictly decreasing runs are distinct.

Original entry on oeis.org

1, 1, 1, 3, 5, 7, 13, 27, 45, 73, 117, 205, 365, 631, 1061, 1711, 2777, 4599, 7657, 12855, 21409, 35059, 56721, 91149, 146161, 234981, 379277, 612825, 988781, 1587635, 2533029, 4017951, 6342853, 9985087, 15699577, 24679859, 38803005, 60979839, 95698257, 149836255

Offset: 0

Gus Wiseman, Jul 29 2024

nonn

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

			The composition (3,1,4,3,2,1,2,8) has strictly decreasing runs ((3,1),(4,3,2,1),(2),(8)), with leaders (3,4,2,8), so is counted under a(24).
The a(0) = 1 through a(6) = 13 compositions:
  ()  (1)  (2)  (3)   (4)    (5)    (6)
                (12)  (13)   (14)   (15)
                (21)  (31)   (23)   (24)
                      (121)  (32)   (42)
                      (211)  (41)   (51)
                             (131)  (123)
                             (311)  (132)
                                    (141)
                                    (213)
                                    (231)
                                    (312)
                                    (321)
                                    (411)

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

For leaders of identical runs we have A274174, ranked by A374249.
The weak opposite version is A374632, ranks A374768.
The opposite version is A374687, ranks A374698.
For identical instead of distinct leaders we have A374760, ranks A374759.
The weak version is A374743, ranks A374701.
Ranked by A374767.
For partitions instead of compositions we have A375133.
Other types of runs:
- For leaders of identical runs we have A000005 for n > 0, ranks A272919.
- For leaders of anti-runs we have A374518, ranked by A374638.
Other types of run-leaders:
- For strictly increasing leaders we have A374762.
- For strictly decreasing leaders we have A374763.
- For weakly increasing leaders we have A374764.
- For weakly decreasing leaders we have A374765.
A003242 counts anti-run compositions, ranks A333489.
A011782 counts compositions.
A238130, A238279, A333755 count compositions by number of runs.
A373949 counts compositions by run-compressed sum, opposite A373951.
A374700 counts compositions by sum of leaders of strictly increasing runs.
Cf. A034296, A106356, A188920, A189076, A238343, A333213, A374517, A374631, A374640, A374686, A374742.

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

dfs(m, r, v) = 1 + sum(s=r, m, if(!setsearch(v, s), dfs(m-s, s, setunion(v, [s]))*x^s + sum(t=1, min(s-1, m-s), dfs(m-s-t, t, setunion(v, [s]))*x^(s+t)*prod(i=t+1, s-1, 1+x^i))));
lista(nn) = Vec(dfs(nn, 1, []) + O(x^(1+nn))); \\ Jinyuan Wang, Feb 13 2025

More terms from Jinyuan Wang, Feb 13 2025

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

Original entry on oeis.org

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

A349053 Number of non-weakly alternating integer compositions of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 4, 12, 37, 95, 232, 533, 1198, 2613, 5619, 11915, 25011, 52064, 107694, 221558, 453850, 926309, 1884942, 3825968, 7749312, 15667596, 31628516, 63766109, 128415848, 258365323, 519392582, 1043405306, 2094829709, 4203577778, 8431313237, 16904555958

Offset: 0

Gus Wiseman, Dec 16 2021

nonn

We define a sequence to be weakly alternating if it is alternately weakly increasing and weakly decreasing, starting with either. Then a sequence is (strongly) alternating iff it is a weakly alternating anti-run.

			The a(6) = 12 compositions:
  (1,1,2,2,1)  (1,1,2,3)  (1,2,4)
  (1,2,1,1,2)  (1,2,3,1)  (4,2,1)
  (1,2,2,1,1)  (1,3,2,1)
  (2,1,1,2,1)  (2,1,1,3)
               (3,1,1,2)
               (3,2,1,1)

Andrew Howroyd, Table of n, a(n) for n = 0..1000 (terms 0..55 from Martin Ehrenstein)
Wikipedia, Alternating permutation

Complementary directed versions are A129852/A129853, strong A025048/A025049.
The strong version is A345192.
The complement is counted by A349052.
These compositions are ranked by A349057, strong A345168.
The complementary version for patterns is A349058, strong A345194.
The complementary multiplicative version is A349059, strong A348610.
An unordered version (partitions) is A349061, complement A349060.
The version for ordered prime factorizations is A349797, complement A349056.
The version for patterns is A350138, strong A350252.
The version for ordered factorizations is A350139.
A001250 counts alternating permutations, complement A348615.
A001700 counts compositions of 2n with alternating sum 0.
A003242 counts Carlitz (anti-run) compositions.
A011782 counts compositions, unordered A000041.
A025047 counts alternating compositions, ranked by A345167.
A106356 counts compositions by number of maximal anti-runs.
A344604 counts alternating compositions with twins.
A345164 counts alternating ordered prime factorizations.
A349054 counts strict alternating compositions.
Cf. A102726, A114901, A128761, A261983, A333213, A333755, A344614, A344615, A345165, A345170, A345195, A349799, A349800, A350251, A350252.

wwkQ[y_]:=And@@Table[If[EvenQ[m],y[[m]]<=y[[m+1]],y[[m]]>=y[[m+1]]],{m,1,Length[y]-1}]||And@@Table[If[EvenQ[m],y[[m]]>=y[[m+1]],y[[m]]<=y[[m+1]]],{m,1,Length[y]-1}];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],!wwkQ[#]&]],{n,0,10}]

a(n) = A011782(n) - A349052(n).

a(21)-a(35) from Martin Ehrenstein, Jan 08 2022

A358836 Number of multiset partitions of integer partitions of n with all distinct block sizes.

Original entry on oeis.org

1, 1, 2, 4, 8, 15, 28, 51, 92, 164, 289, 504, 871, 1493, 2539, 4290, 7201, 12017, 19939, 32911, 54044, 88330, 143709, 232817, 375640, 603755, 966816, 1542776, 2453536, 3889338, 6146126, 9683279, 15211881, 23830271, 37230720, 58015116, 90174847, 139820368, 216286593

Offset: 0

Gus Wiseman, Dec 05 2022

nonn

Also the number of integer compositions of n whose leaders of maximal weakly decreasing runs are strictly increasing. For example, the composition (1,2,2,1,3,1,4,1) has maximal weakly decreasing runs ((1),(2,2,1),(3,1),(4,1)), with leaders (1,2,3,4), so is counted under a(15). - Gus Wiseman, Aug 21 2024

			The a(1) = 1 through a(5) = 15 multiset partitions:
  {1}  {2}    {3}        {4}          {5}
       {1,1}  {1,2}      {1,3}        {1,4}
              {1,1,1}    {2,2}        {2,3}
              {1},{1,1}  {1,1,2}      {1,1,3}
                         {1,1,1,1}    {1,2,2}
                         {1},{1,2}    {1,1,1,2}
                         {2},{1,1}    {1},{1,3}
                         {1},{1,1,1}  {1},{2,2}
                                      {2},{1,2}
                                      {3},{1,1}
                                      {1,1,1,1,1}
                                      {1},{1,1,2}
                                      {2},{1,1,1}
                                      {1},{1,1,1,1}
                                      {1,1},{1,1,1}
From _Gus Wiseman_, Aug 21 2024: (Start)
The a(0) = 1 through a(5) = 15 compositions whose leaders of maximal weakly decreasing runs are strictly increasing:
  ()  (1)  (2)   (3)    (4)     (5)
           (11)  (12)   (13)    (14)
                 (21)   (22)    (23)
                 (111)  (31)    (32)
                        (112)   (41)
                        (121)   (113)
                        (211)   (122)
                        (1111)  (131)
                                (221)
                                (311)
                                (1112)
                                (1121)
                                (1211)
                                (2111)
                                (11111)
(End)

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Gus Wiseman, Sequences counting and ranking compositions by their leaders (for six types of runs).

The version for set partitions is A007837.
For sums instead of sizes we have A271619.
For constant instead of distinct sizes we have A319066.
These multiset partitions are ranked by A326533.
For odd instead of distinct sizes we have A356932.
The version for twice-partitions is A358830.
The case of distinct sums also is A358832.
Ranked by positions of strictly increasing rows in A374740, opposite A374629.
A001970 counts multiset partitions of integer partitions.
A011782 counts compositions.
A063834 counts twice-partitions, strict A296122.
A238130, A238279, A333755 count compositions by number of runs.
A335456 counts patterns matched by compositions.
Runs and leaders: A000041, A188900, A374634, A374635, A374637, A374679, A374742 (ranks A374744), A374743 (ranks A374701), A374746, A374747.
Cf. A000009, A141199, A273873, A279375, A279785, A279790, A358334.
Cf. A003242, A106356, A188920, A189076, A238343, A333213, A336342.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
Table[Length[Select[Join@@mps/@IntegerPartitions[n],UnsameQ@@Length/@#&]],{n,0,10}]
(* second program *)
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], Less@@First/@Split[#,GreaterEqual]&]],{n,0,15}] (* Gus Wiseman, Aug 21 2024 *)

P(n,y) = {1/prod(k=1, n, 1 - y*x^k + O(x*x^n))}
seq(n) = {my(g=P(n,y)); Vec(prod(k=1, n, 1 + polcoef(g, k, y) + O(x*x^n)))} \\ Andrew Howroyd, Dec 31 2022

G.f.: Product_{k>=1} (1 + [y^k]P(x,y)) where P(x,y) = 1/Product_{k>=1} (1 - y*x^k). - Andrew Howroyd, Dec 31 2022

Terms a(11) and beyond from Andrew Howroyd, Dec 31 2022

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

A374634 Number of integer compositions of n whose leaders of weakly increasing runs are strictly increasing.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 12, 17, 28, 43, 67, 103, 162, 245, 374, 569, 854, 1278, 1902, 2816, 4148, 6087, 8881, 12926, 18726, 27042, 38894, 55789, 79733, 113632, 161426, 228696, 323049, 455135, 639479, 896249, 1252905, 1747327, 2431035, 3374603, 4673880, 6459435, 8908173

Offset: 0

Gus Wiseman, Jul 23 2024

nonn

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

			The composition (1,3,3,2,4,3) has weakly increasing runs ((1,3,3),(2,4),(3)), with leaders (1,2,3), so is counted under a(16).
The a(0) = 1 through a(7) = 17 compositions:
  ()  (1)  (2)   (3)    (4)     (5)      (6)       (7)
           (11)  (12)   (13)    (14)     (15)      (16)
                 (111)  (22)    (23)     (24)      (25)
                        (112)   (113)    (33)      (34)
                        (1111)  (122)    (114)     (115)
                                (1112)   (123)     (124)
                                (11111)  (132)     (133)
                                         (222)     (142)
                                         (1113)    (223)
                                         (1122)    (1114)
                                         (11112)   (1123)
                                         (111111)  (1132)
                                                   (1222)
                                                   (11113)
                                                   (11122)
                                                   (111112)
                                                   (1111111)

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

Ranked by positions of strictly increasing rows in A374629 (sums A374630).
Types of runs (instead of weakly increasing):
- For leaders of constant runs we have A000041.
- For leaders of anti-runs we have A374679.
- For leaders of strictly increasing runs we have A374688.
- For leaders of strictly decreasing runs we have A374762.
Types of run-leaders (instead of strictly increasing):
- For strictly decreasing leaders we appear to have A188920.
- For weakly decreasing leaders we appear to have A189076.
- For identical leaders we have A374631.
- For distinct leaders we have A374632, ranks A374768.
- For weakly increasing leaders we have A374635.
A003242 counts anti-run compositions.
A011782 counts compositions.
A238130, A238279, A333755 count compositions by number of runs.
A274174 counts contiguous compositions, ranks A374249.
A335456 counts patterns matched by compositions.
A335548 counts non-contiguous compositions, ranks A374253.
A374637 counts compositions by sum of leaders of weakly increasing runs.
Cf. A106356, A124766, A238343, A261982, A333213, A344604, A373949, A374518, A374687, A374698.

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

dfs(m, r, u) = 1 + sum(s=u+1, min(m, r-1), x^s/(1-x^s) + sum(t=s+1, m-s, dfs(m-s-t, t, s)*x^(s+t)/prod(i=s, t, 1-x^i)));
lista(nn) = Vec(dfs(nn, nn+1, 0) + O(x^(1+nn))); \\ Jinyuan Wang, Feb 13 2025

More terms from Jinyuan Wang, Feb 13 2025

A374687 Number of integer compositions of n whose leaders of strictly increasing runs are distinct.

Original entry on oeis.org

1, 1, 1, 3, 3, 7, 11, 15, 27, 45, 65, 101, 161, 251, 381, 573, 865, 1321, 1975, 2965, 4387, 6467, 9579, 14091, 20669, 30135, 43869, 63531, 91831, 132575, 190567, 273209, 390659, 557069, 792371, 1124381, 1591977, 2249029, 3169993, 4458163, 6256201, 8762251, 12246541

Offset: 0

Gus Wiseman, Jul 27 2024

nonn

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

			The a(0) = 1 through a(7) = 15 compositions:
  ()  (1)  (2)  (3)   (4)   (5)    (6)    (7)
                (12)  (13)  (14)   (15)   (16)
                (21)  (31)  (23)   (24)   (25)
                            (32)   (42)   (34)
                            (41)   (51)   (43)
                            (122)  (123)  (52)
                            (212)  (132)  (61)
                                   (213)  (124)
                                   (231)  (133)
                                   (312)  (142)
                                   (321)  (214)
                                          (241)
                                          (313)
                                          (412)
                                          (421)

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

Ranked by A374698.
Types of runs (instead of strictly increasing):
- For leaders of identical runs we have A274174 for n > 0, ranks A374249.
- For leaders of anti-runs we have A374518, ranks A374638.
- For leaders of weakly increasing runs we have A374632, ranks A374768.
- For leaders of weakly decreasing runs we have A374743, ranks A374701.
- For leaders of strictly decreasing runs we have A374761, ranks A374767.
Types of run-leaders (instead of distinct):
- For identical leaders we have A374686, ranks A374685.
- For strictly increasing leaders we have A374688.
- For strictly decreasing leaders we have A374689.
- For weakly increasing leaders we have A374690.
- For weakly decreasing leaders we have A374697.
A003242 counts anti-run compositions, ranks A333489.
A011782 counts compositions.
A238130, A238279, A333755 count compositions by number of runs.
A335456 counts patterns matched by compositions.
A373949 counts compositions by run-compressed sum, opposite A373951.
A374683 lists leaders of strictly increasing runs of standard compositions.
A374700 counts compositions by sum of leaders of strictly increasing runs.
Cf. A000009, A106356, A188920, A189076, A238343, A333213, A335548, A374629, A374634, A374635.

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

dfs(m, r, v) = 1 + sum(s=1, min(m, r), if(!setsearch(v, s), dfs(m-s, s, setunion(v, [s]))*x^s + sum(t=s+1, m-s, dfs(m-s-t, t, setunion(v, [s]))*x^(s+t)*prod(i=s+1, t-1, 1+x^i))));
lista(nn) = Vec(dfs(nn, nn, []) + O(x^(1+nn))); \\ Jinyuan Wang, Feb 13 2025

More terms from Jinyuan Wang, Feb 13 2025

A345919 Numbers k such that the k-th composition in standard order (row k of A066099) has alternating sum < 0.

Original entry on oeis.org

6, 12, 20, 24, 25, 27, 30, 40, 48, 49, 51, 54, 60, 72, 80, 81, 83, 86, 92, 96, 97, 98, 99, 101, 102, 103, 106, 108, 109, 111, 116, 120, 121, 123, 126, 144, 160, 161, 163, 166, 172, 184, 192, 193, 194, 195, 197, 198, 199, 202, 204, 205, 207, 212, 216, 217, 219

Offset: 1

Gus Wiseman, Jul 09 2021

nonn

The alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i.

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 initial terms and the corresponding compositions:
      6: (1,2)         81: (2,4,1)
     12: (1,3)         83: (2,3,1,1)
     20: (2,3)         86: (2,2,1,2)
     24: (1,4)         92: (2,1,1,3)
     25: (1,3,1)       96: (1,6)
     27: (1,2,1,1)     97: (1,5,1)
     30: (1,1,1,2)     98: (1,4,2)
     40: (2,4)         99: (1,4,1,1)
     48: (1,5)        101: (1,3,2,1)
     49: (1,4,1)      102: (1,3,1,2)
     51: (1,3,1,1)    103: (1,3,1,1,1)
     54: (1,2,1,2)    106: (1,2,2,2)
     60: (1,1,1,3)    108: (1,2,1,3)
     72: (3,4)        109: (1,2,1,2,1)
     80: (2,5)        111: (1,2,1,1,1,1)

The version for Heinz numbers of partitions is A119899.
These are the positions of terms < 0 in A124754.
These compositions are counted by A294175 (even bisection: A008549).
The complement is A345913.
The weak (k <= 0) version is A345915.
The opposite (k < 0) version is A345917.
The version for reversed alternating sum is A345920.
A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.
A011782 counts compositions.
A097805 counts compositions by alternating (or reverse-alternating) sum.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A236913 counts partitions of 2n with reverse-alternating sum <= 0.
A316524 gives the alternating sum of prime indices (reverse: A344616).
A345197 counts compositions by sum, length, and alternating sum.
Standard compositions: A000120, A066099, A070939, A228351, A124754, A344618.
Compositions of n, 2n, or 2n+1 with alternating/reverse-alternating sum k:
- k = 0:  counted by A088218, ranked by A344619/A344619.
- k = 1:  counted by A000984, ranked by A345909/A345911.
- k = -1: counted by A001791, ranked by A345910/A345912.
- k = 2:  counted by A088218, ranked by A345925/A345922.
- k = -2: counted by A002054, ranked by A345924/A345923.
- k >= 0: counted by A116406, ranked by A345913/A345914.
- k <= 0: counted by A058622(n-1), ranked by A345915/A345916.
- k > 0:  counted by A027306, ranked by A345917/A345918.
- k < 0:  counted by A294175, ranked by A345919/A345920.
- k != 0: counted by A058622, ranked by A345921/A345921.
- k even: counted by A081294, ranked by A053754/A053754.
- k odd:  counted by A000302, ranked by A053738/A053738.
Cf. A000070, A000346, A025047, A028260, A032443, A034871, A106356, A114121, A163493, A344608, A344610, A344611, A345908.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
ats[y_]:=Sum[(-1)^(i-1)*y[[i]],{i,Length[y]}];
Select[Range[0,100],ats[stc[#]]<0&]

A345923 Numbers k such that the k-th composition in standard order (row k of A066099) has reverse-alternating sum -2.

Original entry on oeis.org

9, 34, 39, 45, 49, 57, 132, 139, 142, 149, 154, 159, 161, 169, 178, 183, 189, 194, 199, 205, 209, 217, 226, 231, 237, 241, 249, 520, 531, 534, 540, 549, 554, 559, 564, 571, 574, 577, 585, 594, 599, 605, 612, 619, 622, 629, 634, 639, 642, 647, 653, 657, 665

Offset: 1

Gus Wiseman, Jul 10 2021

nonn

The reverse-alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(k-i) y_i.

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 initial terms and the corresponding compositions:
      9: (3,1)            183: (2,1,2,1,1,1)
     34: (4,2)            189: (2,1,1,1,2,1)
     39: (3,1,1,1)        194: (1,5,2)
     45: (2,1,2,1)        199: (1,4,1,1,1)
     49: (1,4,1)          205: (1,3,1,2,1)
     57: (1,1,3,1)        209: (1,2,4,1)
    132: (5,3)            217: (1,2,1,3,1)
    139: (4,2,1,1)        226: (1,1,4,2)
    142: (4,1,1,2)        231: (1,1,3,1,1,1)
    149: (3,2,2,1)        237: (1,1,2,1,2,1)
    154: (3,1,2,2)        241: (1,1,1,4,1)
    159: (3,1,1,1,1,1)    249: (1,1,1,1,3,1)
    161: (2,5,1)          520: (6,4)
    169: (2,2,3,1)        531: (5,3,1,1)
    178: (2,1,3,2)        534: (5,2,1,2)

These compositions are counted by A088218.
These are the positions of 2's in A344618.
The case of partitions of 2n is A344741.
The opposite (negative 2) version is A345923.
The version for unreversed alternating sum is A345925.
The version for Heinz numbers of partitions is A345961.
A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.
A011782 counts compositions.
A097805 counts compositions by alternating (or reverse-alternating) sum.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A120452 counts partitions of 2n with reverse-alternating sum 2.
A316524 gives the alternating sum of prime indices (reverse: A344616).
A344610 counts partitions by sum and positive reverse-alternating sum.
A344611 counts partitions of 2n with reverse-alternating sum >= 0.
A345197 counts compositions by sum, length, and alternating sum.
Standard compositions: A000120, A066099, A070939, A228351, A124754, A344618.
Compositions of n, 2n, or 2n+1 with alternating/reverse-alternating sum k:
- k = 0:  counted by A088218, ranked by A344619/A344619.
- k = 1:  counted by A000984, ranked by A345909/A345911.
- k = -1: counted by A001791, ranked by A345910/A345912.
- k = 2:  counted by A088218, ranked by A345925/A345922.
- k = -2: counted by A002054, ranked by A345924/A345923.
- k >= 0: counted by A116406, ranked by A345913/A345914.
- k <= 0: counted by A058622(n-1), ranked by A345915/A345916.
- k > 0:  counted by A027306, ranked by A345917/A345918.
- k < 0:  counted by A294175, ranked by A345919/A345920.
- k != 0: counted by A058622, ranked by A345921/A345921.
- k even: counted by A081294, ranked by A053754/A053754.
- k odd:  counted by A000302, ranked by A053738/A053738.
Cf. A000070, A000097, A000346, A025047, A032443, A034871, A106356, A163493, A236913, A344607, A344608, A344743.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
sats[y_]:=Sum[(-1)^(i-Length[y])*y[[i]],{i,Length[y]}];
Select[Range[0,100],sats[stc[#]]==-2&]

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A349053 Number of non-weakly alternating integer compositions of n.

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A358836 Number of multiset partitions of integer partitions of n with all distinct block sizes.

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

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A374634 Number of integer compositions of n whose leaders of weakly increasing runs are strictly increasing.

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