A374249 -id:A374249 - 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}]

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]&]

A374765 Number of integer compositions of n whose leaders of strictly decreasing runs are weakly decreasing.

Original entry on oeis.org

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 88, 141, 225, 357, 565, 891, 1399, 2191, 3420, 5321, 8256, 12774, 19711, 30339, 46584, 71359, 109066, 166340, 253163, 384539, 582972, 882166, 1332538, 2009377, 3024969, 4546562, 6822926, 10223632, 15297051, 22855872, 34103117

Offset: 0

Gus Wiseman, Jul 30 2024

nonn

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.

			The composition (3,1,2,2,1) has strictly decreasing runs ((3,1),(2),(2,1)), with leaders (3,2,2), so is counted under a(9).
The a(0) = 1 through a(6) = 13 compositions:
  ()  (1)  (2)   (3)    (4)     (5)      (6)
           (11)  (21)   (22)    (32)     (33)
                 (111)  (31)    (41)     (42)
                        (211)   (212)    (51)
                        (1111)  (221)    (222)
                                (311)    (312)
                                (2111)   (321)
                                (11111)  (411)
                                         (2121)
                                         (2211)
                                         (3111)
                                         (21111)
                                         (111111)

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

The opposite version is A374690.
Other types of runs (instead of strictly decreasing):
- For leaders of identical runs we have A000041.
- For leaders of weakly increasing runs we appear to have A189076.
- For leaders of anti-runs we have A374682.
- For leaders of strictly increasing runs we have A374697.
- For leaders of weakly decreasing runs we have A374747.
Other types of run-leaders (instead of weakly decreasing):
- For identical leaders we have A374760, ranks A374759.
- For distinct leaders we have A374761, ranks A374767.
- For strictly increasing leaders we have A374762.
- For strictly decreasing leaders we have A374763.
- For weakly increasing leaders we have A374764.
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.
A373949 counts compositions by run-compressed sum, opposite A373951.
Cf. A106356, A188900, A188920, A238343, A261982, A333213, A374635, A374636, A374689, A374742, A374743, A375133.

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

dfs(m, r, u) = 1 + sum(s=r, min(m, u), dfs(m-s, s, s)*x^s + sum(t=1, min(s-1, m-s), dfs(m-s-t, t, s)*x^(s+t)*prod(i=t+1, s-1, 1+x^i)));
lista(nn) = Vec(dfs(nn, 1, nn) + O(x^(1+nn))); \\ Jinyuan Wang, Feb 13 2025

More terms from Jinyuan Wang, Feb 13 2025

A376309 Run-lengths of the sequence of first differences of prime-powers.

Original entry on oeis.org

3, 1, 2, 2, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 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, 2, 1, 1, 1, 2, 1, 2, 1, 1, 2, 1, 1, 1, 2

Offset: 1

Gus Wiseman, Sep 22 2024

nonn

			The sequence of prime-powers (A246655) is:
  2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 37, ...
The sequence of first differences (A057820) of prime-powers is:
  1, 1, 1, 2, 1, 1, 2, 2, 3, 1, 2, 4, 2, 2, 2, 2, 1, 5, 4, 2, 4, 2, 4, ...
with runs:
  (1,1,1),(2),(1,1),(2,2),(3),(1),(2),(4),(2,2,2,2),(1),(5),(4),(2),(4), ...
with lengths A376309 (this sequence).

Antti Karttunen, Table of n, a(n) for n = 1..20000

For runs of prime-powers increasing by one we have A174965.
For primes instead of prime-powers we have A333254.
For squarefree numbers instead of prime-powers we have A376306.
For compression instead of run-lengths we have A376308.
For run-sums instead of run-lengths we have A376310.
For positions of first appearances we have A376341, sorted A376340.
A000040 lists the prime numbers, differences A001223.
A000961 and A246655 list prime-powers, first differences A057820.
A003242 counts compressed compositions, ranks A333489.
A005117 lists squarefree numbers, differences A076259.
A024619 and A361102 list non-prime-powers, first differences A375708.
A116861 counts partitions by compressed sum, by compressed length A116608.
A124767 counts runs in standard compositions, anti-runs A333381.
A238130, A238279, A333755 count compositions by number of runs.
A274174 counts contiguous compositions, ranks A374249.
A373948 encodes compression using compositions in standard order.
Cf. A001597, A006549, A025475, A037201, A053289, A110969, A120430, A251092, A373822, A375706.

Length/@Split[Differences[Select[Range[100],PrimePowerQ]]]

up_to = 20000;
A376309list(up_to) = { my(v=vector(up_to), ppp=2, pd=1, d, rl=0, k=2, i=0); while(i<#v, k++; if(isprimepower(k), d = k-ppp; ppp = k; if(d == pd, rl++, i++; v[i] = rl; rl = 1; pd = d))); (v); };
v376309 = A376309list(up_to);
A376309(n) = v376309[n]; \\ Antti Karttunen, Jan 18 2025

More terms from Antti Karttunen, Jan 18 2025

A376310 Run-sums of the sequence of first differences of prime-powers.

Original entry on oeis.org

3, 2, 2, 4, 3, 1, 2, 4, 8, 1, 5, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 6, 4, 6, 8, 4, 2, 4, 2, 4, 8, 4, 2, 1, 3, 6, 2, 10, 2, 12, 4, 2, 4, 6, 2, 10, 2, 4, 2, 24, 4, 2, 4, 6, 4, 8, 5, 1, 12, 2, 6, 4, 2, 6, 4, 14, 4, 2, 4, 14, 12, 4, 2, 4, 6, 2, 18, 4, 6, 8, 4, 8, 10, 2

Offset: 1

Gus Wiseman, Sep 22 2024

nonn

			The sequence of prime-powers (A246655) is:
  2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 37, ...
The sequence of first differences (A057820) of prime-powers is:
  1, 1, 1, 2, 1, 1, 2, 2, 3, 1, 2, 4, 2, 2, 2, 2, 1, 5, 4, 2, 4, 2, 4, ...
with runs:
  (1,1,1),(2),(1,1),(2,2),(3),(1),(2),(4),(2,2,2,2),(1),(5),(4),(2),(4), ...
with sums A376310 (this sequence).

For primes instead of prime-powers we have A373822, halved A373823.
For squarefree numbers instead of prime-powers we have A376307.
For compression instead of run-sums we have A376308.
For run-lengths instead of run-sums we have A376309.
For positions of first appearances we have A376341, sorted A376340.
A000040 lists the prime numbers, differences A001223.
A000961 and A246655 list prime-powers, first differences A057820.
A003242 counts compressed compositions, ranks A333489.
A005117 lists squarefree numbers, differences A076259.
A024619 and A361102 list non-prime-powers, first differences A375708.
A116861 counts partitions by compressed sum, by compressed length A116608.
A124767 counts runs in standard compositions, anti-runs A333381.
A238130, A238279, A333755 count compositions by number of runs.
A274174 counts contiguous compositions, ranks A374249.
A373948 encodes compression using compositions in standard order.
Cf. A001597, A006549, A007916, A025475, A037201, A053289, A054265, A110969, A120430, A174965, A251092, A375706.

Total/@Split[Differences[Select[Range[100],PrimePowerQ]]]

A376342 Positions of 1's in the run-compression (A376305) of the first differences (A076259) of the squarefree numbers (A005117).

Original entry on oeis.org

1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 22, 24, 26, 28, 30, 32, 34, 36, 38, 41, 43, 45, 47, 49, 51, 54, 56, 58, 60, 62, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 100, 103, 105, 107, 109, 111, 113, 115, 117, 119, 121, 124, 126, 128, 130

Offset: 1

Gus Wiseman, Sep 24 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, we can remove all parts equal to the part immediately to their left. For example, (1,1,2,2,1) has run-compression (1,2,1).

			The sequence of squarefree numbers (A005117) is:
  1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 30, ...
with first differences (A076259):
  1, 1, 2, 1, 1, 3, 1, 2, 1, 1, 2, 2, 2, 1, 1, 3, 3, 1, 1, 2, 1, 1, 2, 1, ...
with run-compression (A376305):
  1, 2, 1, 3, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 2, 1, 3, 1, 4, 2, 1, 2, 1, ...
with ones at (A376342):
  1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 22, 24, 26, 28, 30, 32, 34, 36, 38, ...

Before compressing we had A076259.
Positions of 1's in A376305.
The version for nonsquarefree numbers gives positions of ones in A376312.
For prime instead of squarefree numbers we have A376343.
A000040 lists the prime numbers, differences A001223.
A000961 and A246655 list prime-powers, differences A057820.
A003242 counts compressed compositions, ranks A333489.
A005117 lists squarefree numbers, differences A076259 (ones A375927).
A013929 lists nonsquarefree numbers, differences A078147.
A116861 counts partitions by compressed sum, by compressed length A116608.
A274174 counts contiguous compositions, ranks A374249.
Cf. A007674, A037201, A053797, A053806, A061398, A072284, A120992, A373198, A376306, A376307, A376308, A376311.

Join@@Position[First /@ Split[Differences[Select[Range[100],SquareFreeQ]]],1]

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}]

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

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}]

A374639 Numbers k such that the leaders of maximal anti-runs in the k-th composition in standard order (A066099) are not distinct.

Original entry on oeis.org

3, 7, 10, 14, 15, 21, 23, 27, 28, 29, 30, 31, 36, 39, 42, 43, 47, 51, 55, 56, 57, 58, 59, 60, 61, 62, 63, 71, 73, 79, 84, 85, 86, 87, 90, 94, 95, 99, 103, 106, 107, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 135

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 of terms together with the corresponding compositions begins:
   3: (1,1)
   7: (1,1,1)
  10: (2,2)
  14: (1,1,2)
  15: (1,1,1,1)
  21: (2,2,1)
  23: (2,1,1,1)
  27: (1,2,1,1)
  28: (1,1,3)
  29: (1,1,2,1)
  30: (1,1,1,2)
  31: (1,1,1,1,1)

First differs from A335466 in lacking 166, complement A335467.
The complement for leaders of identical runs is A374249, counted by A274174.
For leaders of identical runs we have A374253, counted by A335548.
Positions of non-distinct (or non-strict) rows in A374515.
The complement is A374638, counted by A374518.
For identical instead of non-distinct we have A374519, counted by A374517.
For identical instead of distinct we have A374520, counted by A374640.
Compositions of this type are counted by A374678.
Other functional neighbors are A374768, A374698, A374701, A374767.
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.
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, A114994, A228351, A233564, A238343, A272919, A335466, A373949.

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

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A374765 Number of integer compositions of n whose leaders of strictly decreasing runs are weakly decreasing.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A376309 Run-lengths of the sequence of first differences of prime-powers.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A376310 Run-sums of the sequence of first differences of prime-powers.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

A376342 Positions of 1's in the run-compression (A376305) of the first differences (A076259) of the squarefree numbers (A005117).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

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.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica