A374685 -id:A374685 - cp's OEIS frontend

A374684 Sum of leaders of strictly increasing runs in the n-th composition in standard order.

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

Offset: 0

Gus Wiseman, Jul 26 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 maximal strictly increasing subsequences of the 1234567th composition in standard order are ((3),(2),(1,2),(2),(1,2,5),(1),(1),(1)) with leaders (3,2,1,2,1,1,1,1), so a(1234567) = 12.

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

The weak version is A374630.
Row-sums of A374683.
The opposite version is A374758.
All of the following pertain to compositions in standard order:
- Length is A000120.
- Sum is A029837(n+1) (or sometimes A070939).
- Parts are listed by A066099.
- Number of adjacent equal pairs is A124762, unequal A333382.
- Number of max runs: A124765, A124766, A124767, A124768, A124769, A333381.
- Run-length transform is A333627.
- Run-compression transform is A373948.
- Ranks of contiguous compositions are A374249, counted by A274174.
- Ranks of non-contiguous compositions are A374253, counted by A335548.
Cf. A065120, A106356, A188920, A189076, A238343, A272919, A374634, A374685, A374698.
Cf. A374251 (sums A373953), A374515 (sums A374516), A374740 (sums A374741).

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

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

A374704 Number of ways to choose an integer partition of each part of an integer composition of n (A055887) such that the minima are identical.

Original entry on oeis.org

1, 1, 3, 6, 15, 31, 77, 171, 410, 957, 2275, 5370, 12795, 30366, 72307, 172071, 409875, 976155, 2325804, 5541230, 13204161, 31464226, 74980838, 178684715, 425830008, 1014816979, 2418489344, 5763712776, 13736075563, 32735874251, 78016456122, 185929792353, 443110675075

Offset: 0

Gus Wiseman, Aug 04 2024

nonn

			The a(0) = 1 through a(4) = 15 ways:
  ()  ((1))  ((2))      ((3))          ((4))
             ((1,1))    ((1,2))        ((1,3))
             ((1),(1))  ((1,1,1))      ((2,2))
                        ((1),(1,1))    ((1,1,2))
                        ((1,1),(1))    ((2),(2))
                        ((1),(1),(1))  ((1,1,1,1))
                                       ((1),(1,2))
                                       ((1,2),(1))
                                       ((1),(1,1,1))
                                       ((1,1),(1,1))
                                       ((1,1,1),(1))
                                       ((1),(1),(1,1))
                                       ((1),(1,1),(1))
                                       ((1,1),(1),(1))
                                       ((1),(1),(1),(1))

Andrew Howroyd, Table of n, a(n) for n = 0..1000

A variation for weakly increasing lengths is A141199.
For identical sums instead of minima we have A279787.
The case of reversed twice-partitions is A306319, distinct A358830.
For maxima instead of minima, or for unreversed partitions, we have A358905.
The strict case is A374686 (ranks A374685), maxima A374760 (ranks A374759).
A003242 counts anti-run compositions, ranks A333489.
A011782 counts compositions.
A238130, A238279, A333755 count compositions by number of runs.
A274174 counts contiguous compositions, ranks A374249.
A055887 counts sequences of partitions with total sum n.
A281145 counts same-trees.
A319169 counts partitions with constant Omega, ranked by A320324.
A358911 counts compositions with constant Omega, distinct A358912.
Cf. A000041, A063834, A106356, A189076, A238343, A304969, A305551, A319066, A323429, A333213, A358833, A358835.

Table[Length[Select[Join@@Table[Tuples[IntegerPartitions/@y], {y,Join@@Permutations/@IntegerPartitions[n]}],SameQ@@Min/@#&]],{n,0,15}]

seq(n) = Vec(1 + sum(k=1, n, -1 + 1/(1 - x^k/prod(j=k, n-k, 1 - x^j, 1 + O(x^(n-k+1)))))) \\ Andrew Howroyd, Dec 29 2024

G.f.: 1 + Sum_{k>=1} (-1 + 1/(1 - x^k/Product_{j>=k} (1 - x^j))). - Andrew Howroyd, Dec 29 2024

a(16) onwards from Andrew Howroyd, Dec 29 2024

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]&]

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A374684 Sum of leaders of strictly increasing runs in the n-th composition in standard order.

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

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A374704 Number of ways to choose an integer partition of each part of an integer composition of n (A055887) such that the minima are identical.

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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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