A374700 -id:A374700 - cp's OEIS frontend

A374640 Number of integer compositions of n whose leaders of maximal anti-runs are not identical.

0, 0, 0, 0, 1, 3, 7, 18, 43, 96, 211, 463, 992, 2112, 4462, 9347, 19495, 40480, 83690, 172478, 354455, 726538, 1486024, 3033644, 6182389, 12580486

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 a(0) = 0 through a(7) = 18 compositions:
  .  .  .  .  (211)  (122)   (411)    (133)
                     (311)   (1122)   (322)
                     (2111)  (1221)   (511)
                             (2112)   (1222)
                             (2211)   (2113)
                             (3111)   (2311)
                             (21111)  (3112)
                                      (3211)
                                      (4111)
                                      (11122)
                                      (11221)
                                      (12211)
                                      (21112)
                                      (21121)
                                      (21211)
                                      (22111)
                                      (31111)
                                      (211111)

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

For partitions instead of compositions we have A239955.
The complement is counted by A374517, ranks A374519.
Compositions of this type are ranked by A374520, complement A374519.
For distinct instead of identical leaders we have A374678, ranks A374639, complement A374518, ranks A374638.
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, A238424, A272919, A333213, A373949, A374515, A374700.

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

A374690 Number of integer compositions of n whose leaders of strictly increasing runs are weakly increasing.

Original entry on oeis.org

1, 1, 2, 3, 6, 10, 19, 34, 63, 115, 211, 387, 710, 1302, 2385, 4372, 8009, 14671, 26867, 49196, 90069, 164884, 301812, 552406, 1011004, 1850209, 3385861, 6195832, 11337470, 20745337, 37959030, 69454669, 127081111, 232517129, 425426211, 778376479, 1424137721

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 composition (1,1,3,2,3,2) has strictly increasing runs ((1),(1,3),(2,3),(2)), with leaders (1,1,2,2), so is counted under a(12).
The a(0) = 1 through a(6) = 19 compositions:
  ()  (1)  (2)   (3)    (4)     (5)      (6)
           (11)  (12)   (13)    (14)     (15)
                 (111)  (22)    (23)     (24)
                        (112)   (113)    (33)
                        (121)   (122)    (114)
                        (1111)  (131)    (123)
                                (1112)   (132)
                                (1121)   (141)
                                (1211)   (222)
                                (11111)  (1113)
                                         (1122)
                                         (1131)
                                         (1212)
                                         (1311)
                                         (11112)
                                         (11121)
                                         (11211)
                                         (12111)
                                         (111111)

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).

Ranked by positions of weakly increasing rows in A374683.
Types of runs (instead of strictly increasing):
- For leaders of identical runs we have A000041.
- For leaders of anti-runs we have A374681.
- For leaders of weakly increasing runs we have A374635.
- For leaders of weakly decreasing runs we have A188900.
- For leaders of strictly decreasing runs we have A374764.
Types of run-leaders (instead of weakly increasing):
- For identical leaders we have A374686, ranks A374685.
- For distinct leaders we have A374687, ranks A374698.
- For strictly increasing leaders we have A374688.
- For strictly decreasing leaders we have A374689.
- 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.
A374700 counts compositions by sum of leaders of strictly increasing runs.
Cf. A000009, A106356, A188920, A189076, A238343, A261982, A333213, A374629, A374630, A374632, A374679.

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

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

A375400 Heinz number of the multiset of minima of maximal anti-runs in the weakly increasing prime indices of n.

Original entry on oeis.org

1, 2, 3, 4, 5, 2, 7, 8, 9, 2, 11, 4, 13, 2, 3, 16, 17, 6, 19, 4, 3, 2, 23, 8, 25, 2, 27, 4, 29, 2, 31, 32, 3, 2, 5, 12, 37, 2, 3, 8, 41, 2, 43, 4, 9, 2, 47, 16, 49, 10, 3, 4, 53, 18, 5, 8, 3, 2, 59, 4, 61, 2, 9, 64, 5, 2, 67, 4, 3, 2, 71, 24, 73, 2, 15, 4, 7

Offset: 1

Gus Wiseman, Aug 17 2024

nonn

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
An anti-run is a sequence with no adjacent equal parts. The minima of maximal anti-runs in a sequence are obtained by splitting it into maximal anti-run subsequences and taking the least term of each.
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The prime indices of 540 are (1,1,2,2,2,3), with maximal anti-runs ((1),(1,2),(2),(2,3)), with minima (1,1,2,2), with Heinz number 36, so a(540) = 36.
The prime indices of 990 are (1,2,2,3,5), with maximal anti-runs ((1,2),(2,3,5)), with minima (1,2), with Heinz number 6, so a(990) = 6.

bigomega is A001222(a(n)) = A375136(n).
Least prime factor is A020639(a(n)) = A020639(n).
Least prime index is A055396(a(n)) = A055396(n).
Heinz weights are A056239(a(n)) = A374706(n).
The greatest prime index A061395(a(n)) is the maximum of row n of A375128.
Firsts for omega (except first term) are half A061742.
Prime indices A112798(a(n)) are row n of A375128.
Positions of prime-powers are A375396, counted by A115029.
Positions of squarefree numbers are A375398, counted by A375134.
A000041 counts integer partitions, strict A000009.
A027748 lists distinct prime factors, sum A008472.
A304038 lists distinct prime indices, sum A066328.
A number's prime factors (A027746, reverse A238689) have sum A001414, min A020639, max A006530.
A number's prime indices (A112798, reverse A296150) have sum A056239, min A055396, max A061395.
Both have length A001222, distinct A001221.
Cf. A000040, A046660, A124768, A320324, A358836, A374683, A374700, A375133.

Table[Times@@Prime/@If[n==1,{},Min /@ Split[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]],UnsameQ]],{n,100}]

A374521 Triangle read by rows where T(n,k) is the number of integer compositions of n whose leaders of anti-runs sum to k.

Original entry on oeis.org

1, 0, 1, 0, 0, 2, 0, 1, 1, 2, 0, 2, 1, 2, 3, 0, 2, 5, 3, 4, 2, 0, 5, 7, 8, 3, 5, 4, 0, 9, 12, 11, 17, 5, 8, 2, 0, 14, 26, 23, 22, 24, 6, 9, 4, 0, 25, 42, 54, 41, 36, 36, 7, 12, 3, 0, 46, 76, 88, 107, 60, 60, 48, 9, 14, 4

Offset: 0

Gus Wiseman, Aug 02 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.

			Triangle begins:
   1
   0   1
   0   0   2
   0   1   1   2
   0   2   1   2   3
   0   2   5   3   4   2
   0   5   7   8   3   5   4
   0   9  12  11  17   5   8   2
   0  14  26  23  22  24   6   9   4
   0  25  42  54  41  36  36   7  12   3
   0  46  76  88 107  60  60  48   9  14   4
   0  78 144 166 179 176 101  83  68  10  17   2
   0 136 258 327 339 311 299 139 122  81  12  18   6
   0 242 457 602 704 591 544 447 198 165 109  12  23   2
Row n = 6 counts the following compositions:
  .  (15)    (24)    (321)    (42)     (51)     (6)
     (141)   (114)   (312)    (1122)   (411)    (33)
     (132)   (231)   (1113)   (11112)  (3111)   (222)
     (123)   (213)   (2112)            (2211)   (111111)
     (1212)  (1311)  (1221)            (21111)
             (1131)  (12111)
             (2121)  (11211)
                     (11121)

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

Column n = k is A000005, except a(0) = 1.
Row-sums are A011782.
Column k = 1 is A096569.
For length instead of sum we have A106356.
The corresponding rank statistic is A374516, row-sums of A374515.
For identical leaders we have A374517, ranks A374519.
For distinct leaders we have A374518, ranks A374638.
Other types of runs (instead of anti-):
- For leaders of identical runs we have A373949.
- For leaders of weakly increasing runs we have A374637.
- For leaders of strictly increasing runs we have A374700.
- For leaders of weakly decreasing runs we have A374748.
- For leaders of strictly decreasing runs we have A374766.
A003242 counts anti-run compositions.
A238130, A238279, A333755 count compositions by number of runs.
A274174 counts contiguous compositions, ranks A374249.
Cf. A124766, A238343, A261982, A333213, A374251, A374687, A374761.

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

A374766 Triangle read by rows where T(n,k) is the number of integer compositions of n whose leaders of maximal strictly decreasing runs sum to k.

Original entry on oeis.org

1, 0, 1, 0, 0, 2, 0, 0, 1, 3, 0, 0, 0, 3, 5, 0, 0, 0, 1, 8, 7, 0, 0, 0, 1, 3, 17, 11, 0, 0, 0, 0, 4, 10, 35, 15, 0, 0, 0, 0, 1, 12, 28, 65, 22, 0, 0, 0, 0, 1, 6, 31, 70, 118, 30, 0, 0, 0, 0, 1, 3, 22, 78, 163, 203, 42, 0, 0, 0, 0, 0, 4, 13, 69, 186, 354, 342, 56

Offset: 0

Gus Wiseman, Aug 02 2024

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.

Are the column-sums finite?

			Triangle begins:
   1
   0   1
   0   0   2
   0   0   1   3
   0   0   0   3   5
   0   0   0   1   8   7
   0   0   0   1   3  17  11
   0   0   0   0   4  10  35  15
   0   0   0   0   1  12  28  65  22
   0   0   0   0   1   6  31  70 118  30
   0   0   0   0   1   3  22  78 163 203  42
   0   0   0   0   0   4  13  69 186 354 342  56
Row n = 6 counts the following compositions:
  .  .  .  (321)  (42)    (51)     (6)
                  (132)   (411)    (15)
                  (2121)  (141)    (24)
                          (312)    (114)
                          (231)    (33)
                          (213)    (123)
                          (3111)   (1113)
                          (1311)   (222)
                          (1131)   (1122)
                          (2211)   (11112)
                          (2112)   (111111)
                          (1221)
                          (1212)
                          (21111)
                          (12111)
                          (11211)
                          (11121)

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

Column n = k is A000041.
Row-sums are A011782.
For length instead of sum we have A333213.
The corresponding rank statistic is A374758, row-sums of A374757.
For identical leaders we have A374760, ranks A374759.
For distinct leaders we have A374761, ranks A374767.
Other types of runs (instead of strictly decreasing):
- For leaders of identical runs we have A373949.
- For leaders of anti-runs we have A374521.
- For leaders of weakly increasing runs we have A374637.
- For leaders of strictly increasing runs we have A374700.
- For leaders of weakly decreasing runs we have A374748.
A003242 counts anti-run compositions.
A238130, A238279, A333755 count compositions by number of runs.
A335456 counts patterns matched by compositions.
Cf. A106356, A238343, A261982, A274174, A374517, A374518, A374687.

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

A375135 Number of integer compositions of n whose leaders of maximal strictly increasing runs are not weakly decreasing.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 3, 9, 25, 63, 152, 355, 809, 1804, 3963, 8590, 18423, 39161, 82620, 173198, 361101, 749326, 1548609, 3189132, 6547190, 13404613, 27378579, 55801506, 113517749, 230544752, 467519136, 946815630, 1915199736, 3869892105, 7812086380, 15756526347

Offset: 0

Gus Wiseman, Aug 06 2024

nonn

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

			The composition y = (1,2,1,3,2,3) has strictly increasing runs ((1,2),(1,3),(2,3)), with leaders (1,1,2), which are not weakly decreasing, so y is counted under a(12).
The a(0) = 0 through a(8) = 25 compositions:
  .  .  .  .  .  (122)  (132)   (133)    (143)
                        (1122)  (142)    (152)
                        (1221)  (1132)   (233)
                                (1222)   (1133)
                                (1321)   (1142)
                                (2122)   (1223)
                                (11122)  (1232)
                                (11221)  (1322)
                                (12211)  (1331)
                                         (1421)
                                         (2132)
                                         (3122)
                                         (11132)
                                         (11222)
                                         (11321)
                                         (12122)
                                         (12212)
                                         (12221)
                                         (13211)
                                         (21122)
                                         (21221)
                                         (111122)
                                         (111221)
                                         (112211)
                                         (122111)

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

For leaders of constant runs we have A056823.
For leaders of weakly increasing runs we have A374636, complement A189076?
The complement is counted by A374697.
For leaders of anti-runs we have A374699, complement A374682.
Other functional neighbors: A188920, A374764, A374765.
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. A106356, A238343, A261982, A333213, A374632, A374679, A374683, A374689.

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

a(n) = A011782(n) - A374697(n). - Jinyuan Wang, Feb 13 2025

More terms from Jinyuan Wang, Feb 13 2025

A374699 Number of integer compositions of n whose leaders of maximal anti-runs are not weakly decreasing.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 2, 5, 14, 34, 78, 180, 407, 907, 2000, 4364, 9448, 20323, 43448, 92400, 195604, 412355, 866085, 1813035, 3783895, 7875552

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 a(0) = 0 through a(8) = 14 compositions:
  .  .  .  .  .  (122)  (1122)  (133)    (233)
                        (1221)  (1222)   (1133)
                                (11122)  (1223)
                                (11221)  (1322)
                                (12211)  (1331)
                                         (11222)
                                         (12122)
                                         (12212)
                                         (12221)
                                         (21122)
                                         (111122)
                                         (111221)
                                         (112211)
                                         (122111)

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

The complement is counted by A374682.
Other types of runs (instead of anti-):
- For leaders of identical runs we have A056823.
- For leaders of weakly increasing runs we have A374636, complement A189076?
- For leaders of strictly increasing runs: A375135, complement A374697.
Other types of run-leaders (instead of weakly decreasing):
- For identical leaders we have A374640, ranks A374520, complement A374517, ranks A374519.
- For distinct leaders we have A374678, ranks A374639, complement A374518, ranks A374638.
- For weakly increasing leaders we have complement A374681.
- For strictly increasing leaders we have complement complement A374679.
- For strictly decreasing leaders we have complement 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.
A333381 counts maximal anti-runs in standard compositions.
Cf. A115029, A238343, A333213, A373949, A374515, A374632, A374635, A374700.

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

A375139 Numbers k such that the leaders of strictly increasing runs in the k-th composition in standard order are not weakly decreasing.

Original entry on oeis.org

26, 50, 53, 58, 90, 98, 100, 101, 106, 107, 114, 117, 122, 154, 164, 178, 181, 186, 194, 196, 197, 201, 202, 203, 210, 212, 213, 214, 215, 218, 226, 228, 229, 234, 235, 242, 245, 250, 282, 306, 309, 314, 324, 329, 346, 354, 356, 357, 362, 363, 370, 373, 378

Offset: 1

Gus Wiseman, Aug 12 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 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:
   26: (1,2,2)
   50: (1,3,2)
   53: (1,2,2,1)
   58: (1,1,2,2)
   90: (2,1,2,2)
   98: (1,4,2)
  100: (1,3,3)
  101: (1,3,2,1)
  106: (1,2,2,2)
  107: (1,2,2,1,1)
  114: (1,1,3,2)
  117: (1,1,2,2,1)
  122: (1,1,1,2,2)
  154: (3,1,2,2)
  164: (2,3,3)
  178: (2,1,3,2)
  181: (2,1,2,2,1)
  186: (2,1,1,2,2)

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

For leaders of identical runs we have A335485.
Ranked by positions of non-weakly decreasing rows in A374683.
For identical leaders we have A374685, counted by A374686.
The complement is counted by A374697.
For distinct leaders we have A374698, counted by A374687.
Compositions of this type are counted by A375135.
Weakly increasing leaders: A375137, counts A374636, complement A189076.
Interchanging weak/strict: A375295, counted by A375140, complement A188920.
A003242 counts anti-run compositions, ranks A333489.
A374700 counts compositions by sum of leaders of strictly increasing runs.
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.
- Strict compositions are A233564.
- Run-length transform is A333627, sum A070939.
- Run-compression transform is A373948, sum A373953, excess A373954.
- Run-count: A124766, A124765, A124768, A124769, A333381, A124767.
- Run-leaders: A374629, A374740, A374683, A374757, A374515, A374251.
- Run-rank: A375123, A375124, A375125, A375126, A375127.
Cf. A000009, A261982, A333213, A374520, A374630, A374699, A374768.

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

A374703 Number of integer compositions of 2n whose leaders of weakly decreasing runs sum to n. Center n = 2*k of the triangle A374748.

Original entry on oeis.org

1, 1, 2, 9, 24, 96, 343, 1242, 4700, 17352, 65995

Offset: 0

Gus Wiseman, Aug 12 2024

The weakly decreasing run-leaders of a sequence are obtained by splitting it into maximal weakly decreasing subsequences and taking the first term of each.

			The a(0) = 1 through a(4) = 24 compositions:
  ()  (11)  (22)   (33)     (44)
            (211)  (321)    (422)
                   (1122)   (431)
                   (1221)   (1133)
                   (3111)   (1322)
                   (11112)  (1331)
                   (11121)  (4211)
                   (11211)  (11132)
                   (12111)  (11321)
                            (13211)
                            (21122)
                            (21221)
                            (22112)
                            (22121)
                            (41111)
                            (111113)
                            (111131)
                            (111311)
                            (113111)
                            (131111)
                            (211112)
                            (211121)
                            (211211)
                            (212111)

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

For reversed partitions we have A364910.
For strictly decreasing runs we have the center of A374700.
Center n = 2*k of the triangle A374748.
A003242 counts anti-run compositions.
A011782 counts integer compositions.
A238130, A238279, A333755 count compositions by number of runs.
A274174 counts contiguous compositions, ranks A374249.
Cf. A000041, A004526, A106356, A124766, A188900, A238343, A261982, A333213.

Table[Length[Select[Join@@Permutations /@ IntegerPartitions[2n],Total[First/@Split[#,GreaterEqual]]==n&]],{n,0,8}]

A374705 Number of integer compositions of n whose leaders of maximal strictly increasing runs sum to 2.

Original entry on oeis.org

0, 0, 2, 0, 2, 3, 4, 7, 8, 14, 17, 27, 33, 48, 63, 84, 112, 147, 191, 248, 322, 409, 527, 666, 845, 1062, 1336, 1666, 2079, 2579, 3190, 3936, 4842, 5933, 7259, 8854, 10768, 13074, 15826, 19120, 23048, 27728, 33279, 39879, 47686, 56916, 67818, 80667, 95777, 113552, 134396

Offset: 0

Gus Wiseman, Aug 12 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) = 0 through a(9) = 14 compositions:
  .  .  (2)   .  (112)  (23)   (24)    (25)    (26)    (27)
        (11)     (121)  (113)  (114)   (115)   (116)   (117)
                        (131)  (141)   (151)   (161)   (171)
                               (1212)  (1123)  (1124)  (234)
                                       (1213)  (1214)  (1125)
                                       (1231)  (1241)  (1134)
                                       (1312)  (1313)  (1215)
                                               (1412)  (1251)
                                                       (1314)
                                                       (1341)
                                                       (1413)
                                                       (1512)
                                                       (12123)
                                                       (12312)

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).

For leaders of weakly decreasing runs we have A004526.
The case of strict compositions is A096749.
For leaders of anti-runs we have column k = 2 of A374521.
Leaders of strictly increasing runs in standard compositions are A374683.
Ranked by positions of 2s in A374684.
Column k = 2 of A374700.
A003242 counts anti-run compositions.
A011782 counts compositions.
A238130, A238279, A333755 count compositions by number of runs.
A274174 counts contiguous compositions, ranks A374249.
Cf. A000009, A096765, A106356, A124766, A238343, A261982, A333213.

Table[Length[Select[Join@@Permutations /@ IntegerPartitions[n],Total[First/@Split[#,Less]]==2&]],{n,0,15}]

seq(n)={my(A=O(x^(n-1)), q=eta(x^2 + A)/eta(x + A)); Vec((q*x/(1 + x))^2 + q*x^2/((1 + x)*(1 + x^2)), -n-1)} \\ Andrew Howroyd, Aug 14 2024

G.f.: (x*Q(x)/(1 + x))^2 + x^2*Q(x)/((1 + x)*(1 + x^2)), where Q(x) is the g.f. of A000009. - Andrew Howroyd, Aug 14 2024

a(26) onwards from Andrew Howroyd, Aug 14 2024

cp's OEIS Frontend

A374640 Number of integer compositions of n whose leaders of maximal anti-runs are not identical.

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A374690 Number of integer compositions of n whose leaders of strictly increasing runs are weakly increasing.

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A375400 Heinz number of the multiset of minima of maximal anti-runs in the weakly increasing prime indices of n.

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A374521 Triangle read by rows where T(n,k) is the number of integer compositions of n whose leaders of anti-runs sum to k.

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A374766 Triangle read by rows where T(n,k) is the number of integer compositions of n whose leaders of maximal strictly decreasing runs sum to k.

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A375135 Number of integer compositions of n whose leaders of maximal strictly increasing runs are not weakly decreasing.

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A374699 Number of integer compositions of n whose leaders of maximal anti-runs are not weakly decreasing.

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A375139 Numbers k such that the leaders of strictly increasing runs in the k-th composition in standard order are not weakly decreasing.

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