A374684 -id:A374684 - 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

A374516 Sum of leaders of maximal anti-runs in the n-th composition in standard order.

Original entry on oeis.org

0, 1, 2, 2, 3, 2, 1, 3, 4, 3, 4, 3, 1, 1, 2, 4, 5, 4, 3, 4, 2, 4, 2, 4, 1, 1, 3, 2, 2, 2, 3, 5, 6, 5, 4, 5, 6, 3, 3, 5, 2, 2, 6, 5, 2, 2, 3, 5, 1, 1, 1, 2, 1, 3, 1, 3, 2, 2, 4, 3, 3, 3, 4, 6, 7, 6, 5, 6, 4, 4, 4, 6, 3, 6, 5, 4, 3, 3, 4, 6, 2, 2, 2, 3, 4, 6, 4

Offset: 0

Gus Wiseman, Jul 31 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 1234567th composition in standard order is (3,2,1,2,2,1,2,5,1,1,1), with maximal anti-runs ((3,2,1,2),(2,1,2,5,1),(1),(1)), so a(1234567) is 3 + 2 + 1 + 1 = 7.

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

For length instead of sum we have A333381.
Row-sums of A374515.
Other types of runs (instead of anti-):
- For identical runs we have A373953, row-sums of A374251.
- For weakly increasing runs we have A374630, row-sums of A374629.
- For strictly increasing runs we have A374684, row-sums of A374683.
- For weakly decreasing runs we have A374741, row-sums of A374740.
- For strictly decreasing runs we have A374758, row-sums of A374757.
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.
- Ranks of contiguous compositions are A374249, counted by A274174.
- Ranks of non-contiguous compositions are A374253, counted by A335548.
Six types of maximal runs:
- Count: A124766, A124765, A124768, A124769, A333381, A124767.
- Rank: A375123, A375124, A375125, A375126, A375127, A373948.
Cf. A029931, A228351, A233564, A238343, A272919, A373949, A374517, A374518, A374519, A374638.

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

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

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

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A374516 Sum of leaders of maximal anti-runs in the n-th composition in standard order.

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

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