A333213 -id:A333213 - cp's OEIS frontend

A374688 Number of integer compositions of n whose leaders of strictly increasing runs are themselves strictly increasing.

1, 1, 1, 2, 2, 4, 5, 7, 11, 16, 21, 31, 45, 63, 87, 122, 170, 238, 328, 449, 616, 844, 1151, 1565, 2121, 2861, 3855, 5183, 6953, 9299, 12407, 16513, 21935, 29078, 38468, 50793, 66935, 88037, 115577, 151473, 198175, 258852, 337560, 439507, 571355, 741631

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.

Also the number of ways to choose a strict integer partition of each part of an integer composition of n (A304969) such that the minima are strictly decreasing.

			The a(0) = 1 through a(9) = 16 compositions:
  ()  (1)  (2)  (3)   (4)   (5)    (6)    (7)    (8)     (9)
                (12)  (13)  (14)   (15)   (16)   (17)    (18)
                            (23)   (24)   (25)   (26)    (27)
                            (122)  (123)  (34)   (35)    (36)
                                   (132)  (124)  (125)   (45)
                                          (133)  (134)   (126)
                                          (142)  (143)   (135)
                                                 (152)   (144)
                                                 (233)   (153)
                                                 (1223)  (162)
                                                 (1232)  (234)
                                                         (243)
                                                         (1224)
                                                         (1233)
                                                         (1242)
                                                         (1323)

Christian Sievers, Table of n, a(n) for n = 0..500
Gus Wiseman, Sequences counting and ranking compositions by their leaders (for six types of runs).

The weak version is A374635.
Ranked by positions of strictly increasing rows in A374683 (sums A374684).
The opposite version is A374763.
Types of runs (instead of strictly increasing):
- For leaders of identical runs we have A000041.
- For leaders of anti-runs we have A374679.
- For leaders of weakly increasing runs we have A374634.
- For leaders of strictly decreasing runs we have A374762.
Types of run-leaders (instead of strictly increasing):
- For identical leaders we have A374686, ranks A374685.
- For distinct leaders we have A374687, ranks A374698.
- 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.
A373949 counts compositions by run-compressed sum, opposite A373951.
A374700 counts compositions by sum of leaders of strictly increasing runs.
Cf. A000009, A106356, A188920, A189076, A238343, A261982, A333213, A374632.

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

a(26) and beyond from Christian Sievers, Aug 08 2024

A374697 Number of integer compositions of n whose leaders of strictly increasing runs are weakly decreasing.

Original entry on oeis.org

1, 1, 2, 4, 8, 15, 29, 55, 103, 193, 360, 669, 1239, 2292, 4229, 7794, 14345, 26375, 48452, 88946, 163187, 299250, 548543, 1005172, 1841418, 3372603, 6175853, 11307358, 20699979, 37890704, 69351776, 126926194, 232283912, 425075191, 777848212, 1423342837, 2604427561

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.

Also the number of ways to choose a strict integer partition of each part of an integer composition of n (A304969) such that the minima are weakly decreasing [weakly increasing works too].

			The composition (1,2,1,3,2,3) has strictly increasing runs ((1,2),(1,3),(2,3)), with leaders (1,1,2), so is not counted under a(12).
The a(0) = 1 through a(5) = 15 compositions:
  ()  (1)  (2)   (3)    (4)     (5)
           (11)  (12)   (13)    (14)
                 (21)   (22)    (23)
                 (111)  (31)    (32)
                        (112)   (41)
                        (121)   (113)
                        (211)   (131)
                        (1111)  (212)
                                (221)
                                (311)
                                (1112)
                                (1121)
                                (1211)
                                (2111)
                                (11111)

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 opposite version is A374764.
Ranked by positions of weakly decreasing rows in A374683.
Interchanging weak/strict appears to give A188920, opposite A358836.
Types of runs (instead of strictly increasing):
- For leaders of identical runs we have A000041.
- For leaders of anti-runs we have A374682.
- For leaders of weakly increasing runs we have A189076, complement A374636.
- For leaders of weakly decreasing runs we have A374747.
- For leaders of strictly decreasing runs we have A374765.
Types of run-leaders (instead of weakly decreasing):
- For identical leaders we have A374686, ranks A374685.
- For distinct leaders we have A374687, ranks A374698.
- For weakly increasing leaders we have A374690.
- For strictly increasing leaders we have A374688.
- For strictly decreasing leaders we have A374689.
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.
A374700 counts compositions by sum of leaders of strictly increasing runs.
Cf. A000009, A106356, A238343, A261982, A333213, A374632, A374679, A374740.

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

seq(n) = Vec(1/prod(k=1, n, 1 - x^k*prod(j=k+1, n-k, 1 + x^j, 1 + O(x^(n-k+1))))) \\ Andrew Howroyd, Jul 31 2024

G.f.: 1/(Product_{k>=1} (1 - x^k*Product_{j>=k+1} (1 + x^j))). - Andrew Howroyd, Jul 31 2024

a(26) onwards from Andrew Howroyd, Jul 31 2024

A124760 Number of rises for compositions in standard order.

Original entry on oeis.org

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

Offset: 0

Franklin T. Adams-Watters, Nov 06 2006

The standard order of compositions is given by A066099.
A114994 seems to give the positions of zeros. - Antti Karttunen, Jul 09 2017
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. a(n) is one fewer than the number of maximal weakly decreasing runs in this composition. Alternatively, a(n) is the number of strict ascents in the same composition. For example, the weakly decreasing runs of the 1234567th composition are ((3,2,1),(2,2,1),(2),(5,1,1,1)), so a(1234567) = 4 - 1 = 3. The 3 strict ascents together with the weak descents are: 3 >= 2 >= 1 < 2 >= 2 >= 1 < 2 < 5 >= 1 >= 1 >= 1. - Gus Wiseman, Apr 08 2020

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

Antti Karttunen, Table of n, a(n) for n = 0..16383

Cf. A066099, A124761, A124762, A124763, A124764, A011782 (row lengths), A045883 (row sums), A233249, A333213, A333380.
Compositions of n with k strict ascents are A238343.
All of the following pertain to compositions in standard order (A066099):
- Length is A000120.
- Sum is A070939.
- Weakly decreasing compositions are A114994.
- 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.
- Weakly increasing compositions are A225620.
- Reverse is A228351 (triangle).
- Strict compositions are A233564.
- Constant compositions are A272919.
- Normal compositions are A333217.
- Strictly decreasing compositions are A333255.
- Strictly increasing compositions are A333256.
- Anti-runs are counted by A333381.
- Adjacent unequal pairs are counted by A333382.
- Anti-runs are A333489.
- Runs-resistance is A333628.

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

A066099row(n) = {my(v=vector(n), j=0, k=0); while(n>0, k++; if(n%2==1, v[j++]=k; k=0); n\=2);  vector(j, i, v[j-i+1]); } \\ Returns empty for n=0. - From code of Franklin T. Adams-Watters in A066099.
A124760(n) = { my(v=A066099row(n), r=0); for(i=2,length(v),r += (v[i]>v[i-1])); (r); }; \\ Antti Karttunen, Jul 09 2017

For a composition b(1),...,b(k), a(n) = Sum_{i = 1 .. k-1} [b(i+1) > b(i)], where [ ] is Iverson bracket, giving in this case 1 only if b(i+1) > b(i), and 0 otherwise. - Formula clarified by Antti Karttunen, Jul 10 2017

For n > 0, a(n) = A124765(n) - 1. - Gus Wiseman, Apr 08 2020

A335471 Number of compositions of n avoiding the pattern (1,2,1).

Original entry on oeis.org

1, 1, 2, 4, 7, 13, 23, 40, 67, 115, 190, 311, 505, 807, 1285, 2031, 3164, 4896, 7550, 11499, 17480, 26379, 39558, 58946, 87469, 129051, 189484, 277143, 403477, 584653, 844236, 1213743, 1738372, 2481770, 3528698, 5003364, 7070225, 9958387, 13982822, 19580613, 27333403

Offset: 0

Gus Wiseman, Jun 17 2020

nonn

Also the number of (1,1,2)-avoiding or (2,1,1)-avoiding compositions.
We define a pattern to be a finite sequence covering an initial interval of positive integers. Patterns are counted by A000670 and ranked by A333217. A sequence S is said to match a pattern P if there is a not necessarily contiguous subsequence of S whose parts have the same relative order as P. For example, (3,1,1,3) matches (1,1,2), (2,1,1), and (2,1,2), but avoids (1,2,1), (1,2,2), and (2,2,1).
A composition of n is a finite sequence of positive integers summing to n.

			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)
                        (211)   (113)
                        (1111)  (122)
                                (212)
                                (221)
                                (311)
                                (1112)
                                (2111)
                                (11111)

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Wikipedia, Permutation pattern
Gus Wiseman, Sequences counting and ranking compositions by the patterns they match or avoid.

The version for patterns is A001710.
The version for prime indices is A335449.
These compositions are ranked by A335467.
The complement A335470 is the matching version.
The (2,1,2)-avoiding version is A335473.
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.
Compositions are counted by A011782.
Compositions avoiding (1,2,3) are counted by A102726.
Non-unimodal compositions are counted by A115981 and ranked by A335373.
Combinatory separations are counted by A269134.
Patterns matched by compositions are counted by A335456.
Minimal patterns avoided by a standard composition are counted by A335465.
Cf. A034691, A056986, A106356, A232464, A238279, A333213, A333755.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],!MatchQ[#,{_,x_,_,y_,_,x_,_}/;x

a(n)={local(Cache=Map()); my(F(n,m,k)=if(m>n, m=n); if(m==0, n==0, my(hk=[n,m,k], z); if(!mapisdefined(Cache,hk,&z), z=self()(n,m-1,k) + k*sum(i=1,n\m, self()(n-i*m, m-1, k+i)); mapput(Cache, hk, z)); z)); F(n,n,1)} \\ Andrew Howroyd, Dec 31 2020

a(n > 0) = 2^(n - 1) - A335470(n).

a(n) = F(n,n,1) where F(n,m,k) = F(n,m-1,k) + k*(Sum_{i=1..floor(n/m)} F(n-i*m, m-1, k+i)) for m > 0 with F(0,m,k)=1 and F(n,0,k)=0 otherwise. - Andrew Howroyd, Dec 31 2020

Terms a(21) and beyond from Andrew Howroyd, Dec 31 2020

A333212 Lengths of maximal weakly decreasing subsequences in the sequence of prime gaps (A001223).

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Mar 14 2020

nonn

Prime gaps are differences between adjacent prime numbers.

			The prime gaps split into the following weakly decreasing subsequences: (1), (2,2), (4,2), (4,2), (4), (6,2), (6,4,2), (4), (6,6,2), (6,4,2), (6,4), (6), ...

First differences of A258025 (with zero prepended).
The version for the Kolakoski sequence is A332273.
The weakly increasing version is A333215.
The unequal version is A333216.
The strictly decreasing version is A333252.
The strictly increasing version is A333253.
The equal version is A333254.
Prime gaps are A001223.
Positions of adjacent equal differences are A064113.
Weakly decreasing runs of compositions in standard order are A124765.
Positions of strict ascents in the sequence of prime gaps are A258025.
Cf. A000040, A000720, A001221, A036263, A054819, A084758, A114994, A124760, A124761, A124768, A333213, A333214.

Length/@Split[Differences[Array[Prime,100]],#1>=#2&]//Most

Ones correspond to weak prime quartets A054819, so the sum of terms up to but not including the n-th one is A000720(A054819(n - 1)).

A374678 Number of integer compositions of n whose leaders of maximal anti-runs are not distinct.

Original entry on oeis.org

0, 0, 1, 1, 3, 7, 15, 32, 70, 144, 311, 653, 1354, 2820, 5850, 12054, 24810, 50923, 104206, 212841, 433919, 882930, 1793810, 3639248, 7373539, 14921986

Offset: 0

Gus Wiseman, Aug 06 2024

The leaders of maximal 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 anti-runs of y = (1,1,2,2) are ((1),(1,2),(2)) with leaders (1,1,2) so y is counted under a(6).
The a(0) = 0 through a(6) = 15 compositions:
  .  .  (11)  (111)  (22)    (113)    (33)
                     (112)   (221)    (114)
                     (1111)  (1112)   (222)
                             (1121)   (1113)
                             (1211)   (1122)
                             (2111)   (1131)
                             (11111)  (1311)
                                      (2211)
                                      (3111)
                                      (11112)
                                      (11121)
                                      (11211)
                                      (12111)
                                      (21111)
                                      (111111)

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

For constant runs we have A335548, complement A274174, ranks A374249.
The complement is counted by A374518, ranks A374638.
For weakly increasing runs we have complement A374632, ranks A374768.
Compositions of this type are ranked by A374639.
For identical instead of distinct leaders we have A374640, ranks A374520, complement A374517, ranks A374519.
A003242 counts anti-runs, ranks A333489.
A065120 gives leaders of standard compositions.
A106356 counts compositions by number of maximal anti-runs.
A238279 counts compositions by number of maximal runs
A274174 counts contiguous compositions, ranks A374249.
Cf. A034296, A188920, A189076, A238343, A239955, A333213, A373949, A374515.

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

A124761 Number of falls for compositions in standard order.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 1, 1, 0, 2, 1, 1, 0, 1, 0, 1, 1, 2, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 1, 1, 1, 2, 1, 1, 0, 1, 1, 2, 1, 2, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 0, 1, 1, 1, 0, 2, 1, 1, 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. a(n) is one fewer than the number of maximal weakly increasing runs in this composition. Alternatively, a(n) is the number of strict descents in the same composition. For example, the weakly increasing of runs of the 1234567th composition are ((3),(2),(1,2,2),(1,2,5),(1,1,1)), so a(1234567) = 5 - 1 = 4. The 4 strict descents together with the weak ascents are: 3 > 2 > 1 <= 2 <= 2 > 1 <= 2 <= 5 > 1 <= 1 <= 1. - 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 0
  0 1 0 0
  0 1 0 1 0 1 0 0
  0 1 1 1 0 1 1 1 0 1 0 1 0 1 0 0
  0 1 1 1 0 2 1 1 0 1 0 1 1 2 1 1 0 1 1 1 0 1 1 1 0 1 0 1 0 1 0 0

Cf. A066099, A124760, A124763, A124764, A011782 (row lengths), A045883 (row sums), A333213, A333220, A333379.
Positions of zeros are A225620.
Compositions of n with k strict descents are A238343.
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.
- Strictly increasing runs are counted by A124768.
- Strictly decreasing runs are counted by A124769.
- Weakly increasing compositions are A225620.
- Reverse is A228351 (triangle).
- Strict compositions are A233564.
- Initial intervals are A246534.
- Constant compositions are A272919.
- Normal compositions are A333217.
- Permutations are A333218.
- Strictly decreasing compositions are A333255.
- Strictly increasing compositions are A333256.
- Runs-resistance is A333628.

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

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

For n > 0, a(n) = A124766(n) - 1. - Gus Wiseman, Apr 08 2020

A374680 Number of integer compositions of n whose leaders of anti-runs are strictly decreasing.

Original entry on oeis.org

1, 1, 1, 3, 5, 8, 16, 31, 52, 98, 179, 323, 590, 1078, 1945, 3531, 6421, 11621, 21041, 38116, 68904, 124562, 225138, 406513, 733710, 1323803

Offset: 0

Gus Wiseman, Aug 01 2024

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) = 16 compositions:
  ()  (1)  (2)  (3)   (4)    (5)    (6)
                (12)  (13)   (14)   (15)
                (21)  (31)   (23)   (24)
                      (121)  (32)   (42)
                      (211)  (41)   (51)
                             (131)  (123)
                             (212)  (132)
                             (311)  (141)
                                    (213)
                                    (231)
                                    (312)
                                    (321)
                                    (411)
                                    (1212)
                                    (2112)
                                    (2121)

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

For distinct but not necessarily decreasing leaders we have A374518.
For partitions instead of compositions we have A375133.
Other types of runs (instead of anti-):
- For leaders of identical runs we have A000041.
- For leaders of weakly increasing runs we have A188920.
- For leaders of weakly decreasing runs we have A374746.
- For leaders of strictly decreasing runs we have A374763.
- For leaders of strictly increasing runs we have A374689.
Other types of run-leaders (instead of strictly decreasing):
- For identical leaders we have A374517, ranks A374519.
- For distinct leaders we have A374518, ranks A374638.
- For weakly increasing leaders we have A374681.
- For strictly increasing leaders we have A374679.
- For weakly decreasing leaders we have A374682.
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. A189076, A238343, A333213, A333381, A373949, A374515, A374632, A374678, A374700, A374706.

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

A374681 Number of integer compositions of n whose leaders of anti-runs are weakly increasing.

Original entry on oeis.org

1, 1, 2, 4, 7, 14, 27, 50, 96, 185, 353, 672, 1289, 2466, 4722, 9052, 17342, 33244, 63767, 122325, 234727, 450553, 864975, 1660951, 3190089, 6128033

Offset: 0

Gus Wiseman, Aug 01 2024

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) = 14 compositions:
  ()  (1)  (2)   (3)    (4)     (5)
           (11)  (12)   (13)    (14)
                 (21)   (22)    (23)
                 (111)  (31)    (32)
                        (112)   (41)
                        (121)   (113)
                        (1111)  (122)
                                (131)
                                (212)
                                (221)
                                (1112)
                                (1121)
                                (1211)
                                (11111)

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

For partitions instead of compositions we have A034296.
Other types of runs (instead of anti-):
- For leaders of constant runs we have A000041.
- For leaders of weakly decreasing runs we have A188900.
- For leaders of weakly increasing runs we have A374635.
- For leaders of strictly increasing runs we have A374690.
- For leaders of strictly decreasing runs we have A374764.
Other types of run-leaders (instead of weakly increasing):
- For identical leaders we have A374517, ranks A374519.
- For distinct leaders we have A374518, ranks A374638.
- 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. A189076, A238343, A333213, A333381, A373949, A374515, A374632, A374678, A374700, A374706.

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

A374682 Number of integer compositions of n whose leaders of anti-runs are weakly decreasing.

Original entry on oeis.org

1, 1, 2, 4, 8, 15, 30, 59, 114, 222, 434, 844, 1641, 3189, 6192, 12020, 23320, 45213, 87624, 169744, 328684, 636221, 1231067, 2381269, 4604713, 8901664

Offset: 0

Gus Wiseman, Aug 01 2024

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) = 15 compositions:
  ()  (1)  (2)   (3)    (4)     (5)
           (11)  (12)   (13)    (14)
                 (21)   (22)    (23)
                 (111)  (31)    (32)
                        (112)   (41)
                        (121)   (113)
                        (211)   (131)
                        (1111)  (212)
                                (221)
                                (311)
                                (1112)
                                (1121)
                                (1211)
                                (2111)
                                (11111)

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

For reversed partitions instead of compositions we have A115029.
The complement is A374699.
Other types of runs (instead of anti-):
- For leaders of identical runs we have A000041.
- For leaders of weakly increasing runs we have A189076, complement A374636.
- For leaders of weakly decreasing runs we have A374747.
- For leaders of strictly decreasing runs we have A374765.
- For leaders of strictly increasing runs we have A374697.
Other types of run-leaders (instead of weakly decreasing):
- For identical leaders we have A374517, ranks A374519.
- For distinct leaders we have A374518, ranks A374638.
- For weakly increasing leaders we have A374681.
- For strictly increasing leaders we have A374679.
- 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. A238343, A333213, A333381, A373949, A374515, A374632, A374635, A374678, A374700, A374706.

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

cp's OEIS Frontend

A374688 Number of integer compositions of n whose leaders of strictly increasing runs are themselves strictly increasing.

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A374697 Number of integer compositions of n whose leaders of strictly increasing runs are weakly decreasing.

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A124760 Number of rises for compositions in standard order.

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A335471 Number of compositions of n avoiding the pattern (1,2,1).

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A333212 Lengths of maximal weakly decreasing subsequences in the sequence of prime gaps (A001223).

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A374678 Number of integer compositions of n whose leaders of maximal anti-runs are not distinct.

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A124761 Number of falls for compositions in standard order.

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