A374251 -id:A374251 - 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}]

A375137 Numbers k such that the k-th composition in standard order (row k of A066099) matches the dashed pattern 1-32.

Original entry on oeis.org

50, 98, 101, 114, 178, 194, 196, 197, 202, 203, 210, 226, 229, 242, 306, 324, 354, 357, 370, 386, 388, 389, 393, 394, 395, 402, 404, 405, 406, 407, 418, 421, 434, 450, 452, 453, 458, 459, 466, 482, 485, 498, 562, 610, 613, 626, 644, 649, 690, 706, 708, 709

Offset: 1

Gus Wiseman, Aug 09 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.
These are also numbers k such that the maximal weakly increasing runs in the k-th composition in standard order do not have weakly decreasing leaders, where 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 reverse version (A375138) ranks compositions matching the dashed pattern 23-1.

			Composition 102 is (1,3,1,2), which matches 1-3-2 but not 1-32.
Composition 210 is (1,2,3,2), which matches 1-32 but not 132.
Composition 358 is (2,1,3,1,2), which matches 2-3-1 and 1-3-2 but not 23-1 or 1-32.
The terms together with corresponding compositions begin:
   50: (1,3,2)
   98: (1,4,2)
  101: (1,3,2,1)
  114: (1,1,3,2)
  178: (2,1,3,2)
  194: (1,5,2)
  196: (1,4,3)
  197: (1,4,2,1)
  202: (1,3,2,2)
  203: (1,3,2,1,1)
  210: (1,2,3,2)
  226: (1,1,4,2)
  229: (1,1,3,2,1)
  242: (1,1,1,3,2)

Wikipedia, Permutation pattern.

The complement is too dense, but counted by A189076.
The non-dashed version is A335480, reverse A335482.
For leaders of identical runs we have A335485, reverse A335486.
For identical leaders we have A374633, counted by A374631.
Compositions of this type are counted by A374636.
For distinct leaders we have A374768, counted by A374632.
The reverse version is A375138, counted by A374636.
For leaders of strictly increasing runs we have A375139, counted by A375135.
Matching 1-21 also gives A375295, counted by A375140 (complement A188920).
A003242 counts anti-runs, ranks A333489.
A011782 counts compositions.
A238130, A238279, A333755 count compositions by number of 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, reverse A228351.
- Number of adjacent equal pairs is A124762, unequal A333382.
- Strict compositions are A233564.
- Constant compositions are A272919.
- Run-length transform is A333627, sum A070939.
- Run-counts: A124766, A124765, A124768, A124769, A333381, A124767.
- Run-leaders: A374629, A374740, A374683, A374757, A374515, A374251.
Cf. A056823, A106356, A188919, A238343, A333213, A373948, A373953, A374634, A374635, A374637, A375123, A375296.

stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,100],MatchQ[stc[#],{_,x_,_,z_,y_,_}/;x

A374520 Numbers k such that the leaders of maximal anti-runs in the k-th composition in standard order (A066099) are not identical.

Original entry on oeis.org

11, 19, 23, 26, 35, 39, 43, 46, 47, 53, 58, 67, 71, 74, 75, 78, 79, 83, 87, 91, 92, 93, 94, 95, 100, 106, 107, 117, 122, 131, 135, 138, 139, 142, 143, 147, 149, 151, 154, 155, 156, 157, 158, 159, 163, 164, 167, 171, 174, 175, 179, 183, 184, 185, 186, 187, 188

Offset: 1

Gus Wiseman, Aug 06 2024

nonn

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 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 sequence together with corresponding compositions begins:
  11: (2,1,1)
  19: (3,1,1)
  23: (2,1,1,1)
  26: (1,2,2)
  35: (4,1,1)
  39: (3,1,1,1)
  43: (2,2,1,1)
  46: (2,1,1,2)
  47: (2,1,1,1,1)
  53: (1,2,2,1)
  58: (1,1,2,2)
  67: (5,1,1)
  71: (4,1,1,1)
  74: (3,2,2)
  75: (3,2,1,1)
  78: (3,1,1,2)
  79: (3,1,1,1,1)
  83: (2,3,1,1)
  87: (2,2,1,1,1)
  91: (2,1,2,1,1)

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

For leaders of maximal constant runs we have the complement of A272919.
Positions of non-constant rows in A374515.
The complement is A374519, counted by A374517.
For distinct instead of identical leaders we have A374639, counted by A374678, complement A374638, counted by A374518.
Compositions of this type are counted by A374640.
A065120 gives leaders of standard compositions.
A106356 counts compositions by number of maximal anti-runs.
A238279 counts compositions by number of maximal runs
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.
Six types of maximal runs:
- Count: A124766, A124765, A124768, A124769, A333381, A124767.
- Leaders: A374629, A374740, A374683, A374757, A374515, A374251.
- Rank: A375123, A375124, A375125, A375126, A375127, A373948.
Cf. A029931, A228351, A233564, A238343, A238424, A333213, A373949.

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

A374250 Greatest sum of run-compression of a permutation of the prime factors of n.

Original entry on oeis.org

0, 2, 3, 2, 5, 5, 7, 2, 3, 7, 11, 7, 13, 9, 8, 2, 17, 8, 19, 9, 10, 13, 23, 7, 5, 15, 3, 11, 29, 10, 31, 2, 14, 19, 12, 10, 37, 21, 16, 9, 41, 12, 43, 15, 11, 25, 47, 7, 7, 12, 20, 17, 53, 8, 16, 11, 22, 31, 59, 12, 61, 33, 13, 2, 18, 16, 67, 21, 26, 14, 71

Offset: 1

Gus Wiseman, Jul 09 2024

nonn

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 prime factors of 24 are {2,2,2,3}, with permutations such as (2,2,3,2) whose run-compression sums to 7, so a(24) = 7.
The prime factors of 216 are {2,2,2,3,3,3}, with permutations such as (2,3,2,3,2,3) whose run-compression sums to 15, so a(216) = 15.

Positions of 2 are A000079 (powers of two) except 1.
Positions of 3 are A000244 (powers of three) except 1.
For least instead of greatest sum of run-compression we have A008472.
For prime indices instead of factors we have A373956.
For number of runs instead of sum of run-compression we have A373957.
A001221 counts distinct prime factors, A001222 with multiplicity.
A003242 counts run-compressed compositions, i.e., anti-runs.
A007947 (squarefree kernel) represents run-compression of multisets.
A008480 counts permutations of prime factors (or prime indices).
A056239 adds up prime indices, row sums of A112798.
A116861 counts partitions by sum of run-compression.
A304038 lists run-compression of prime indices, sum A066328.
A335433 lists numbers whose prime indices are separable, complement A335448.
A373949 counts compositions by sum of run-compression, opposite A373951.
A374251 run-compresses standard compositions, sum A373953, rank A373948.
Cf. A000040, A001223, A001414, A037201, A116608, A124767, A333755, A373954, A374246, A374247, A374252.

prifacs[n_]:=If[n==1,{},Flatten[ConstantArray@@@FactorInteger[n]]];
Table[Max@@(Total[First/@Split[#]]& /@ Permutations[prifacs[n]]),{n,100}]

a(n) = A001414(n) iff n belongs to A335433 (the separable case, complement A335448), row-sums of A027746.

A373956 Greatest sum of run-compression of a permutation of the prime indices of n.

Original entry on oeis.org

0, 1, 2, 1, 3, 3, 4, 1, 2, 4, 5, 4, 6, 5, 5, 1, 7, 5, 8, 5, 6, 6, 9, 4, 3, 7, 2, 6, 10, 6, 11, 1, 7, 8, 7, 6, 12, 9, 8, 5, 13, 7, 14, 7, 7, 10, 15, 4, 4, 7, 9, 8, 16, 5, 8, 6, 10, 11, 17, 7, 18, 12, 8, 1, 9, 8, 19, 9, 11, 8, 20, 7, 21, 13, 8, 10, 9, 9, 22, 5

Offset: 1

Gus Wiseman, Jul 06 2024

nonn

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).

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 24 are {1,1,1,2}, with permutations such as (1,1,2,1) whose run-compression sums to 4, so a(24) = 4.
The prime indices of 216 are {1,1,1,2,2,2}, with permutations such as (1,2,1,2,1,2) whose run-compression sums to 9, so a(216) = 9.

Positions of first appearances are 1 followed by the primes A000040.
Positions of 1 are A000079 (powers of two) except 1.
Positions of 2 are A000244 (powers of three) except 1.
Positions of 3 are {6} U A000351 (six or powers of five) except 1.
For number of runs instead of sum of run-compression we have A373957.
For prime factors instead of indices we have A374250.
A001221 counts distinct prime factors, A001222 with multiplicity.
A003242 counts run-compressed compositions, i.e., anti-runs.
A007947 (squarefree kernel) represents run-compression of multisets.
A008480 counts permutations of prime factors (or prime indices).
A056239 adds up prime indices, row sums of A112798.
A116861 counts partitions by sum of run-compression.
A304038 lists run-compression of prime indices, sum A066328.
A335433 lists numbers whose prime indices are separable, complement A335448.
A373949 counts compositions by sum of run-compression, opposite A373951.
A374251 run-compresses standard compositions, sum A373953, rank A373948.
Cf. A001223, A001414, A037201, A116608, A124767, A316413, A333755, A340387, A344415, A373954, A374246, A374247.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Max@@(Total[First/@Split[#]]&/@Permutations[prix[n]]),{n,100}]

a(n) = A056239(n) iff n belongs to A335433 (the separable case), complement A335448.

A374247 The greatest number of runs possible in a permutation of the prime factors of n (A373957) minus the number of distinct such factors (A001221).

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Jul 07 2024

nonn

If n has separable prime factors (A335433), then a(n) = A001222(n) - A001221(n) = A046660(n). A multiset is separable iff it has an anti-run permutation (meaning there are no adjacent equal parts).

			The runs of the 6 permutations of the prime factors of 36 are:
  ((2,2),(3,3))
  ((2),(3),(2),(3))
  ((2),(3,3),(2))
  ((3),(2,2),(3))
  ((3),(2),(3),(2))
  ((3,3),(2,2))
The longest length is 4, so a(36) = 4 - 2 = 2.

Positions of first appearances appear to be A026549.
Positions of nonzero terms are A126706, complement A303554.
This is an opposite version of A373957.
The sister-sequence A374246 uses A001222 instead of A001221.
This is the number of nonzero terms in row n of A374252.
A003242 counts run-compressed compositions, i.e., anti-runs.
A008480 counts permutations of prime factors, by number of runs A374252.
A027746 lists prime factors, row-sums A001414.
A027748 is run-compression of prime factors, row-sums A008472.
A304038 is run-compression of prime indices, row-sums A066328.
A333755 counts compositions by number of runs.
A335433 lists separable numbers, complement A335448.
A374250 maximizes sum of run-compression, for indices A373956.
Cf. A007947, A046660, A124767, A151821, A238130, A373951, A374251.

prifacs[n_]:=If[n==1,{},Flatten[ConstantArray@@@FactorInteger[n]]];
Table[Max@@Table[Length[Split[y]], {y,Permutations[prifacs[n]]}]-PrimeNu[n],{n,100}]

a(n) = A373957(n) - A001221(n).

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}]

A375138 Numbers k such that the k-th composition in standard order (row k of A066099) matches the dashed pattern 23-1.

Original entry on oeis.org

41, 81, 83, 105, 145, 161, 163, 165, 166, 167, 169, 209, 211, 233, 289, 290, 291, 297, 321, 323, 325, 326, 327, 329, 331, 332, 333, 334, 335, 337, 339, 361, 401, 417, 419, 421, 422, 423, 425, 465, 467, 489, 545, 553, 577, 578, 579, 581, 582, 583, 593, 595, 617

Offset: 1

Gus Wiseman, Aug 09 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.
These are also numbers k such that the maximal weakly increasing runs in the reverse of the k-th composition in standard order do not have weakly decreasing leaders, where 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 reverse version (A375137) ranks compositions matching the dashed pattern 1-32.

			Composition 89 is (2,1,3,1), which matches 2-3-1 but not 23-1.
Composition 165 is (2,3,2,1), which matches 23-1 but not 231.
Composition 358 is (2,1,3,1,2), which matches 2-3-1 and 1-3-2 but not 23-1 or 1-32.
The sequence together with corresponding compositions begins:
   41: (2,3,1)
   81: (2,4,1)
   83: (2,3,1,1)
  105: (1,2,3,1)
  145: (3,4,1)
  161: (2,5,1)
  163: (2,4,1,1)
  165: (2,3,2,1)
  166: (2,3,1,2)
  167: (2,3,1,1,1)
  169: (2,2,3,1)
  209: (1,2,4,1)
  211: (1,2,3,1,1)
  233: (1,1,2,3,1)

Wikipedia, Permutation pattern.

The complement is too dense, but counted by A189076.
The non-dashed version is A335482, reverse A335480.
For leaders of identical runs we have A335486, reverse A335485.
Compositions of this type are counted by A374636.
The reverse version is A375137, counted by A374636.
Matching 12-1 also gives A375296, counted by A375140 (complement A188920).
A003242 counts anti-runs, ranks A333489.
A011782 counts compositions.
A238130, A238279, A333755 count compositions by number of 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, reverse A228351.
- Number of adjacent equal pairs is A124762, unequal A333382.
- Strict compositions are A233564.
- Constant compositions are A272919.
- Run-length transform is A333627, sum A070939.
- Run-counts: A124766, A124765, A124768, A124769, A333381, A124767.
- Run-leaders: A374629, A374740, A374683, A374757, A374515, A374251.
Cf. A056823, A106356, A188919, A238343, A333213, A335466, A373948, A373953, A374633, A375123, A375139, A374768.

stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,100],MatchQ[stc[#],{_,y_,z_,_,x_,_}/;x

A374248 Sum of prime indices of n (with multiplicity) minus the greatest possible sum of run-compression of a permutation of the prime indices of n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Jul 10 2024

nonn

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).

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 96 are {1,1,1,1,1,2}, with sum 7, and we have permutations such as (1,1,1,1,2,1), with run-compression (1,2,1), with sum 4, so a(96) = 7 - 4 = 3.

Positions of zeros are A335433 (separable).
Positions of positive terms are A335448 (inseparable).
This is an opposite version of A373956, for prime factors A374250.
For prime factors instead of indices we have A374255.
A001221 counts distinct prime factors, A001222 with multiplicity.
A003242 counts run-compressed compositions, i.e., anti-runs.
A007947 (squarefree kernel) represents run-compression of multisets.
A008480 counts permutations of prime factors.
A027746 lists prime factors, row-sums A001414.
A027748 is run-compression of prime factors, row-sums A008472.
A056239 adds up prime indices, row-sums of A112798.
A116861 counts partitions by sum of run-compression.
A304038 is run-compression of prime indices, row-sums A066328.
A373949 counts compositions by sum of run-compression, opposite A373951.
A373957 gives greatest number of runs in a permutation of prime factors.
A374251 run-compresses standard compositions, sum A373953, rank A373948.
A374252 counts permutations of prime factors by number of runs.
Cf. A000040, A026549, A037201, A046660, A124767, A280286, A280292, A333755, A373954, A374246, A374247.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Total[prix[n]]-Max@@(Total[First/@Split[#]]&/@Permutations[prix[n]]),{n,100}]

a(n) = A056239(n) - A373956(n).

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}]

cp's OEIS Frontend

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

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A375137 Numbers k such that the k-th composition in standard order (row k of A066099) matches the dashed pattern 1-32.

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A374520 Numbers k such that the leaders of maximal anti-runs in the k-th composition in standard order (A066099) are not identical.

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A374250 Greatest sum of run-compression of a permutation of the prime factors of n.

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A373956 Greatest sum of run-compression of a permutation of the prime indices of n.

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A374247 The greatest number of runs possible in a permutation of the prime factors of n (A373957) minus the number of distinct such factors (A001221).

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

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A375138 Numbers k such that the k-th composition in standard order (row k of A066099) matches the dashed pattern 23-1.

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