A333755 -id:A333755 - cp's OEIS frontend

A374632 Number of integer compositions of n whose leaders of weakly increasing runs are distinct.

1, 1, 2, 4, 7, 13, 23, 40, 69, 119, 200, 335, 557, 917, 1499, 2433, 3920, 6280, 10004, 15837, 24946, 39087, 60952, 94606, 146203, 224957, 344748, 526239, 800251, 1212527, 1830820, 2754993, 4132192, 6178290, 9209308, 13686754, 20282733, 29973869, 44175908, 64936361

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 (4,2,2,1,1,3) has weakly increasing runs ((4),(2,2),(1,1,3)), with leaders (4,2,1), so is counted under a(13).
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)

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

Ranked by A374768 = positions of distinct rows in A374629 (sums A374630).
Types of runs (instead of weakly increasing):
- For leaders of constant runs we have A274174, ranks A374249.
- For leaders of anti-runs we have A374518, ranks A374638.
- For leaders of strictly increasing runs we have A374687, ranks A374698.
- For leaders of weakly decreasing runs we have A374743, ranks A335467.
- For leaders of strictly decreasing runs we have A374761, ranks A374767.
Types of run-leaders (instead of distinct):
- 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 weakly increasing leaders we have A374635.
- For strictly increasing leaders we have A374634.
A003242 counts anti-run compositions.
A011782 counts compositions.
A238130, A238279, A333755 count compositions by number of runs.
A335456 counts patterns matched by compositions.
A374637 counts compositions by sum of leaders of weakly increasing runs.
Cf. A106356, A124766, A238343, A261982, A333213, A335548, A373949, A373953.

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

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

More terms from Jinyuan Wang, Feb 13 2025

A374635 Number of integer compositions of n whose leaders of weakly increasing runs are themselves weakly increasing.

Original entry on oeis.org

1, 1, 2, 3, 6, 10, 20, 36, 69, 130, 247, 467, 890, 1689, 3213, 6110, 11627, 22121, 42101, 80124, 152512, 290300, 552609, 1051953, 2002583, 3812326, 7257679, 13816867, 26304254, 50077792, 95338234, 181505938, 345554234, 657874081, 1252478707, 2384507463, 4539705261

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 (1,3,3,2,4,2) has weakly increasing runs ((1,3,3),(2,4),(2)), with leaders (1,2,2), so is counted under a(15).
The a(0) = 1 through a(6) = 20 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)
                                         (1221)
                                         (1311)
                                         (11112)
                                         (11121)
                                         (11211)
                                         (12111)
                                         (111111)

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

Ranked by positions of weakly increasing rows in A374629 (sums A374630).
Types of runs (instead of weakly increasing):
- For leaders of constant runs we have A000041.
- For leaders of weakly decreasing runs we have A188900.
- For leaders of anti-runs we have A374681.
- For leaders of strictly increasing runs we have A374690.
- For leaders of strictly decreasing runs we have A374764.
Types of run-leaders (instead of weakly 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 strictly increasing leaders we have A374634.
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, A374518, A374687, A374761.

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

dfs(m, r, u) = 1 + sum(s=u, 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, 1) + O(x^(1+nn))); \\ Jinyuan Wang, Feb 13 2025

More terms from Jinyuan Wang, Feb 13 2025

A353932 Irregular triangle read by rows where row k lists the run-sums of the k-th composition in standard order.

Original entry on oeis.org

1, 2, 2, 3, 2, 1, 1, 2, 3, 4, 3, 1, 4, 2, 2, 1, 3, 1, 2, 1, 2, 2, 4, 5, 4, 1, 3, 2, 3, 2, 2, 3, 4, 1, 2, 1, 2, 2, 3, 1, 4, 1, 3, 1, 1, 4, 1, 2, 2, 2, 3, 2, 2, 1, 3, 2, 5, 6, 5, 1, 4, 2, 4, 2, 6, 3, 2, 1, 3, 1, 2, 3, 3, 2, 4, 2, 3, 1, 6, 4, 2, 2, 1, 3

Offset: 1

Gus Wiseman, Jun 10 2022

Every sequence can be uniquely split into a sequence of non-overlapping runs. For example, the runs of (2,2,1,1,1,3,2,2) are ((2,2),(1,1,1),(3),(2,2)), with sums (4,3,3,4).

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.

			Triangle begins:
  1
  2
  2
  3
  2 1
  1 2
  3
  4
  3 1
  4
  2 2
  1 3
  1 2 1
For example, composition 350 in standard order is (2,2,1,1,1,2), so row 350 is (4,3,2).

Row-sums are A029837.
Standard compositions are listed by A066099.
Row-lengths are A124767.
These compositions are ranked by A353847.
Row k has A353849(k) distinct parts.
The version for partitions is A354584, ranked by A353832.
A005811 counts runs in binary expansion.
A300273 ranks collapsible partitions, counted by A275870.
A353838 ranks partitions with all distinct run-sums, counted by A353837.
A353851 counts compositions with all equal run-sums, ranked by A353848.
A353840-A353846 pertain to partition run-sum trajectory.
A353852 ranks compositions with all distinct run-sums, counted by A353850.
A353853-A353859 pertain to composition run-sum trajectory.
A353860 counts collapsible compositions.
Cf. A003242, A175413, A181819, A238279, A274174, A333381, A333489, A333755, A353835, A353839, A353863, A353864.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Table[Total/@Split[stc[n]],{n,0,30}]

A373954 Excess run-compression of standard compositions. Sum of all parts minus sum of compressed parts of the n-th integer composition in standard order.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 2, 1, 0, 0, 1, 3, 0, 0, 0, 1, 0, 2, 0, 2, 0, 0, 2, 1, 1, 1, 2, 4, 0, 0, 0, 1, 3, 0, 0, 2, 0, 0, 4, 3, 0, 0, 1, 3, 0, 0, 0, 1, 0, 2, 0, 2, 1, 1, 3, 2, 2, 2, 3, 5, 0, 0, 0, 1, 0, 0, 0, 2, 0, 3, 2, 1, 0, 0, 1, 3, 0, 0, 0, 1, 2, 4, 2

Offset: 0

Gus Wiseman, Jun 27 2024

nonn

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.

We define the (run-) compression of a sequence to be the anti-run obtained by reducing each run of repeated parts to a single part. Alternatively, compression removes all parts equal to the part immediately to their left. For example, (1,1,2,2,1) has compression (1,2,1).

			The excess compression of (2,1,1,3) is 1, so a(92) = 1.

Gus Wiseman, Statistics, classes, and transformations of standard compositions

For length instead of sum we have A124762, counted by A106356.
The opposite for length is A124767, counted by A238279 and A333755.
Positions of zeros are A333489, counted by A003242.
Positions of nonzeros are A348612, counted by A131044.
Compositions counted by this statistic are A373951, opposite A373949.
Compression of standard compositions is A373953.
Positions of ones are A373955.
A037201 gives compression of first differences of primes, halved A373947.
A066099 lists the parts of all compositions in standard order.
A114901 counts compositions with no isolated parts.
A116861 counts partitions by this statistic, by length A116608.
A240085 counts compositions with no unique parts.
A333627 takes the rank of a composition to the rank of its run-lengths.
Cf. A070939, A238130, A238343, A272919, A285981, A333213, A333381, A333382.

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

a(n) = A029837(n) - A373953(n).

A353852 Numbers k such that the k-th composition in standard order (row k of A066099) has all distinct run-sums.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 15, 16, 17, 18, 19, 20, 21, 23, 24, 26, 28, 30, 31, 32, 33, 34, 35, 36, 37, 38, 40, 41, 42, 43, 44, 47, 48, 50, 51, 52, 55, 56, 57, 58, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 79, 80, 81, 84, 85, 86, 87, 88

Offset: 0

Gus Wiseman, May 31 2022

nonn

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.

Every sequence can be uniquely split into a sequence of non-overlapping runs. For example, the runs of (2,2,1,1,1,3,2,2) are ((2,2),(1,1,1),(3),(2,2)), with sums (4,3,3,4).

			The terms together with their binary expansions and corresponding compositions begin:
   0:        0  ()
   1:        1  (1)
   2:       10  (2)
   3:       11  (1,1)
   4:      100  (3)
   5:      101  (2,1)
   6:      110  (1,2)
   7:      111  (1,1,1)
   8:     1000  (4)
   9:     1001  (3,1)
  10:     1010  (2,2)
  12:     1100  (1,3)
  15:     1111  (1,1,1,1)
  16:    10000  (5)
  17:    10001  (4,1)
  18:    10010  (3,2)
  19:    10011  (3,1,1)
  20:    10100  (2,3)
  21:    10101  (2,2,1)
  23:    10111  (2,1,1,1)

The version for runs in binary expansion is A175413.
The version for parts instead of run-sums is A233564, counted A032020.
The version for run-lengths instead of run-sums is A351596, counted A329739.
The version for runs instead of run-sums is A351290, counted by A351013.
The version for partitions is A353838, counted A353837, complement A353839.
The equal instead of distinct version is A353848, counted by A353851.
These compositions are counted by A353850.
The weak version (rucksack compositions) is A354581, counted by A354580.
A003242 counts anti-run compositions, ranked by A333489.
A005811 counts runs in binary expansion.
A011782 counts compositions.
A242882 counts composition with distinct multiplicities, partitions A098859.
A304442 counts partitions with all equal run-sums.
A351014 counts distinct runs in standard compositions, firsts A351015.
A353853-A353859 pertain to composition run-sum trajectory.
A353864 counts rucksack partitions, perfect A353865.
A353929 counts distinct runs in binary expansion, firsts A353930.
Cf. A044813, A238279, A333755, A351016, A351017, A353832, A353847, A353849, A353860, A353863, A353932.

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

A344615 Number of compositions of n with no adjacent triples (..., x, y, z, ...) where x <= y <= z.

Original entry on oeis.org

1, 1, 2, 3, 6, 10, 17, 29, 50, 84, 143, 241, 408, 688, 1162, 1959, 3305, 5571, 9393, 15832, 26688, 44980, 75812, 127769, 215338, 362911, 611620, 1030758, 1737131, 2927556, 4933760, 8314754, 14012668, 23615198, 39798098, 67070686, 113032453, 190490542, 321028554

Offset: 0

Gus Wiseman, May 27 2021

nonn

These compositions avoid the weak consecutive pattern (1,2,3), the strict version being A128761.

			The a(1) = 1 through a(6) = 17 compositions:
  (1)  (2)    (3)    (4)      (5)        (6)
       (1,1)  (1,2)  (1,3)    (1,4)      (1,5)
              (2,1)  (2,2)    (2,3)      (2,4)
                     (3,1)    (3,2)      (3,3)
                     (1,2,1)  (4,1)      (4,2)
                     (2,1,1)  (1,3,1)    (5,1)
                              (2,1,2)    (1,3,2)
                              (2,2,1)    (1,4,1)
                              (3,1,1)    (2,1,3)
                              (1,2,1,1)  (2,3,1)
                                         (3,1,2)
                                         (3,2,1)
                                         (4,1,1)
                                         (1,2,1,2)
                                         (1,3,1,1)
                                         (2,1,2,1)
                                         (2,2,1,1)

The case of permutations is A049774.
The strict non-adjacent version is A102726.
The case of permutations of prime indices is A344652.
A001250 counts alternating permutations.
A005649 counts anti-run patterns.
A106356 counts compositions by number of maximal anti-runs.
A114901 counts compositions where each part is adjacent to an equal part.
A344604 counts wiggly compositions with twins.
A344605 counts wiggly patterns with twins.
A344606 counts wiggly permutations of prime factors with twins.
Counting compositions by patterns:
- A003242 avoiding (1,1) adjacent.
- A011782 no conditions.
- A106351 avoiding (1,1) adjacent by sum and length.
- A128695 avoiding (1,1,1) adjacent.
- A128761 avoiding (1,2,3).
- A232432 avoiding (1,1,1).
- A335456 all patterns.
- A335457 all patterns adjacent.
- A335514 matching (1,2,3).
- A344604 weakly avoiding (1,2,3) and (3,2,1) adjacent.
- A344614 avoiding (1,2,3) and (3,2,1) adjacent.
- A344615 weakly avoiding (1,2,3) adjacent.
Cf. A000041, A006330, A008965, A027187, A238279/A333755, A333213, A335464, A335515, A344612, A344619.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],!MatchQ[#,{_,x_,y_,z_,_}/;x<=y<=z]&]],{n,0,15}]

More terms from Bert Dobbelaere, Jun 12 2021

A353851 Number of integer compositions of n with all equal run-sums.

Original entry on oeis.org

1, 1, 2, 2, 5, 2, 8, 2, 12, 5, 8, 2, 34, 2, 8, 8, 43, 2, 52, 2, 70, 8, 8, 2, 282, 5, 8, 18, 214, 2, 386, 2, 520, 8, 8, 8, 1957, 2, 8, 8, 2010, 2, 2978, 2, 3094, 94, 8, 2, 16764, 5, 340, 8, 12310, 2, 26514, 8, 27642, 8, 8, 2, 132938, 2, 8, 238, 107411, 8, 236258

Offset: 0

Gus Wiseman, May 31 2022

Every sequence can be uniquely split into a sequence of non-overlapping runs. For example, the runs of (2,2,1,1,1,3,2,2) are ((2,2),(1,1,1),(3),(2,2)), with sums (4,3,3,4).

			The a(0) = 1 through a(8) = 12 compositions:
  ()  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
           (11)  (111)  (22)    (11111)  (33)      (1111111)  (44)
                        (112)            (222)                (224)
                        (211)            (1113)               (422)
                        (1111)           (2112)               (2222)
                                         (3111)               (11114)
                                         (11211)              (41111)
                                         (111111)             (111122)
                                                              (112112)
                                                              (211211)
                                                              (221111)
                                                              (11111111)
For example:
  (1,1,2,1,1) has run-sums (2,2,2) so is counted under a(6).
  (4,1,1,1,1,2,2) has run-sums (4,4,4) so is counted under a(12).
  (3,3,2,2,2) has run-sums (6,6) so is counted under a(12).

David A. Corneth, Table of n, a(n) for n = 0..10000

The version for parts or runs instead of run-sums is A000005.
The version for multiplicities instead of run-sums is A098504.
All parts are divisors of n, see A100346.
The version for partitions is A304442, ranked by A353833.
The version for run-lengths instead of run-sums is A329738, ptns A047966.
These compositions are ranked by A353848.
The distinct instead of equal version is A353850.
A003242 counts anti-run compositions, ranked by A333489.
A005811 counts runs in binary expansion.
A011782 counts compositions.
A353847 represents the composition run-sum transformation.
For distinct instead of equal run-sums: A032020, A098859, A242882, A329739, A351013, A353837, ranked by A353838 (complement A353839), A353852, A354580, ranked by A354581.
Cf. A000005, A006881, A238279, A275870, A333755, A351014, A351016, A351017, A353832, A353834, A353849, A353853-A353859 (run-sum trajectory), A353860, A353863, A353864, A353932.

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

a(n) = {if(n <=1, return(1)); my(d = divisors(n), res = 0); for(i = 1, #d, nd = numdiv(d[i]); res+=(nd*(nd-1)^(n/d[i]-1)) ); res } \\ David A. Corneth, Jun 02 2022

From David A. Corneth, Jun 02 2022 (Start)
a(p) = 2 for prime p.
a(p*q) = 8 for distinct primes p and q (Cf. A006881).
a(n) = Sum_{d|n} tau(d)*(tau(d)-1) ^ (n/d - 1) where tau = A000005. (End)

More terms from David A. Corneth, Jun 02 2022

A373951 Triangle read by rows where T(n,k) is the number of integer compositions of n such that replacing each run of repeated parts with a single part (run-compression) yields a composition of n - k.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 3, 0, 1, 0, 4, 2, 1, 1, 0, 7, 4, 4, 0, 1, 0, 14, 5, 6, 5, 1, 1, 0, 23, 14, 10, 10, 6, 0, 1, 0, 39, 26, 29, 12, 14, 6, 1, 1, 0, 71, 46, 54, 40, 19, 16, 9, 0, 1, 0, 124, 92, 96, 82, 64, 22, 22, 8, 1, 1, 0, 214, 176, 204, 144, 137, 82, 30, 26, 10, 0, 1, 0

Offset: 0

Gus Wiseman, Jun 28 2024

			Triangle begins:
    1
    1   0
    1   1   0
    3   0   1   0
    4   2   1   1   0
    7   4   4   0   1   0
   14   5   6   5   1   1   0
   23  14  10  10   6   0   1   0
   39  26  29  12  14   6   1   1   0
   71  46  54  40  19  16   9   0   1   0
  124  92  96  82  64  22  22   8   1   1   0
Row n = 6 counts the following compositions:
  (6)     (411)   (3111)   (33)     (222)  (111111)  .
  (51)    (114)   (1113)   (2211)
  (15)    (1311)  (1221)   (1122)
  (42)    (1131)  (12111)  (21111)
  (24)    (2112)  (11211)  (11112)
  (141)           (11121)
  (321)
  (312)
  (231)
  (213)
  (132)
  (123)
  (2121)
  (1212)
For example, the composition (1,2,2,1) with compression (1,2,1) is counted under T(6,2).

Column k = 0 is A003242 (anti-runs or compressed compositions).
Row-sums are A011782.
Same as A373949 with rows reversed.
Column k = 1 is A373950.
This statistic is represented by A373954, difference A373953.
A114901 counts compositions with no isolated parts.
A116861 counts partitions by compressed sum, by compressed length A116608.
A124767 counts runs in standard compositions, anti-runs A333381.
A240085 counts compositions with no unique parts.
A333755 counts compositions by compressed length.
A373948 represents the run-compression transformation.
Cf. A037201 (halved A373947), A106356, A124762, A238130, A238279, A238343, A285981, A333213, A333382, A333489, A373952.

Table[Length[Select[Join@@Permutations /@ IntegerPartitions[n], Total[First/@Split[#]]==n-k&]],{n,0,10},{k,0,n}]

A374251 Irregular triangle read by rows where row n is the run-compression of the n-th composition in standard order.

Original entry on oeis.org

1, 2, 1, 3, 2, 1, 1, 2, 1, 4, 3, 1, 2, 2, 1, 1, 3, 1, 2, 1, 1, 2, 1, 5, 4, 1, 3, 2, 3, 1, 2, 3, 2, 1, 2, 1, 2, 2, 1, 1, 4, 1, 3, 1, 1, 2, 1, 2, 1, 1, 3, 1, 2, 1, 1, 2, 1, 6, 5, 1, 4, 2, 4, 1, 3, 3, 2, 1, 3, 1, 2, 3, 1, 2, 4, 2, 3, 1, 2, 2, 1, 2, 1, 3

Offset: 1

Gus Wiseman, Jul 09 2024

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.

We define the run-compression of a sequence to be the anti-run obtained by reducing each run of repeated parts to a single part. Alternatively, run-compression removes all parts equal to the part immediately to their left. For example, (1,1,2,2,1) has run-compression (1,2,1).

			The standard compositions and their run-compressions begin:
   0: ()        --> ()
   1: (1)       --> (1)
   2: (2)       --> (2)
   3: (1,1)     --> (1)
   4: (3)       --> (3)
   5: (2,1)     --> (2,1)
   6: (1,2)     --> (1,2)
   7: (1,1,1)   --> (1)
   8: (4)       --> (4)
   9: (3,1)     --> (3,1)
  10: (2,2)     --> (2)
  11: (2,1,1)   --> (2,1)
  12: (1,3)     --> (1,3)
  13: (1,2,1)   --> (1,2,1)
  14: (1,1,2)   --> (1,2)
  15: (1,1,1,1) --> (1)

Gus Wiseman, Statistics, classes, and transformations of standard compositions

Last column is A001511.
First column is A065120.
Row-lengths are A124767.
Using prime indices we get A304038, row-sums A066328.
Row n has A334028(n) distinct elements.
Rows are ranked by A373948 (standard order).
Row-sums are A373953.
A003242 counts run-compressed compositions, i.e., anti-runs, ranks A333489.
A007947 (squarefree kernel) represents run-compression of multisets.
A037201 run-compresses first differences of primes, halved A373947.
A066099 lists the parts of compositions in standard order.
A116861 counts partitions by sum of run-compression.
A238279 and A333755 count compositions by number of runs.
A373949 counts compositions by sum of run-compression, opposite A373951.
Cf. A000120, A070939, A106356, A124762, A233564, A238130, A238343, A272919, A333381, A333382, A333627, A373954, A374250.

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

A353849 Number of distinct positive run-sums of the n-th composition in standard order.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 2, 2, 2, 1, 3, 3, 1, 2, 3, 1, 2, 3, 2, 1, 2, 2, 2, 3, 3, 3, 2, 2, 3, 2, 3, 2, 1, 1, 3, 2, 1, 1, 2, 2, 2, 2, 3, 3, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 2, 3

Offset: 0

Gus Wiseman, May 30 2022

nonn

Every sequence can be uniquely split into a sequence of non-overlapping runs. For example, the runs of (2,2,1,1,1,3,2,2) are ((2,2),(1,1,1),(3),(2,2)), with sums (4,3,3,4).

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.

			Composition 462903 in standard order is (1,1,4,7,1,2,1,1,1), with run-sums (2,4,7,1,2,3), of which a(462903) = 5 are distinct.

Mathematics Stack Exchange, What is a sequence run? (answered 2011-12-01)

Counting repeated runs also gives A124767.
Positions of first appearances are A246534.
For distinct runs instead of run-sums we have A351014 (firsts A351015).
A version for partitions is A353835, weak A353861.
Positions of 1's are A353848, counted by A353851.
The version for binary expansion is A353929 (firsts A353930).
The run-sums themselves are listed by A353932, with A353849 distinct terms.
For distinct run-lengths instead of run-sums we have A354579.
A005811 counts runs in binary expansion.
A066099 lists compositions in standard order.
A165413 counts distinct run-lengths in binary expansion.
A297770 counts distinct runs in binary expansion, firsts A350952.
A353847 represents the run-sum transformation for compositions.
A353853-A353859 pertain to composition run-sum trajectory.
Distinct runs: A032020, A175413, A351013, A351018, A329739, A351290.
Selected statistics of standard compositions:
- Length is A000120.
- Sum is A070939.
- Heinz number is A333219.
- Number of distinct parts is A334028.
Selected classes of standard compositions:
- Partitions are A114994, strict A333256.
- Multisets are A225620, strict A333255.
- Strict compositions are A233564.
- Constant compositions are A272919.
Cf. A003242, A044813, A071625, A238279, A329738, A333381, A333489, A333755, A353744, A353832, A353850, A353852, A353866.

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

cp's OEIS Frontend

A374632 Number of integer compositions of n whose leaders of weakly increasing runs are distinct.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A374635 Number of integer compositions of n whose leaders of weakly increasing runs are themselves weakly increasing.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A353932 Irregular triangle read by rows where row k lists the run-sums of the k-th composition in standard order.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A373954 Excess run-compression of standard compositions. Sum of all parts minus sum of compressed parts of the n-th integer composition in standard order.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A353852 Numbers k such that the k-th composition in standard order (row k of A066099) has all distinct run-sums.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A344615 Number of compositions of n with no adjacent triples (..., x, y, z, ...) where x <= y <= z.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Extensions

A353851 Number of integer compositions of n with all equal run-sums.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A373951 Triangle read by rows where T(n,k) is the number of integer compositions of n such that replacing each run of repeated parts with a single part (run-compression) yields a composition of n - k.

Views

Author