cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 16 results. Next

A384175 Number of subsets of {1..n} with all distinct lengths of maximal runs (increasing by 1).

Original entry on oeis.org

1, 2, 4, 7, 13, 24, 44, 77, 135, 236, 412, 713, 1215, 2048, 3434, 5739, 9559, 15850, 26086, 42605, 69133, 111634, 179602, 288069, 460553, 733370, 1162356, 1833371, 2878621, 4501856, 7016844, 10905449, 16904399, 26132460, 40279108, 61885621, 94766071, 144637928
Offset: 0

Views

Author

Gus Wiseman, Jun 16 2025

Keywords

Examples

			The subset {2,3,5,6,7,9} has maximal runs ((2,3),(5,6,7),(9)), with lengths (2,3,1), so is counted under a(9).
The a(0) = 1 through a(4) = 13 subsets:
  {}  {}   {}     {}       {}
      {1}  {1}    {1}      {1}
           {2}    {2}      {2}
           {1,2}  {3}      {3}
                  {1,2}    {4}
                  {2,3}    {1,2}
                  {1,2,3}  {2,3}
                           {3,4}
                           {1,2,3}
                           {1,2,4}
                           {1,3,4}
                           {2,3,4}
                           {1,2,3,4}

Links

Crossrefs

For equal instead of distinct lengths we have A243815.
These subsets are ranked by A328592.
The complement is counted by A384176.
For anti-runs instead of runs we have A384177, ranks A384879.
For partitions instead of subsets we have A384884, A384178, A384886, A384880.
For permutations instead of subsets we have A384891, equal instead of distinct A384892.
A034839 counts subsets by number of maximal runs, for strict partitions A116674.
A098859 counts Wilf partitions (distinct multiplicities), complement A336866.
A384893 counts subsets by number of maximal anti-runs, for partitions A268193, A384905.
Cf. A000009, A010027, A044813, A047993, A242882, A325325, A329739, A351202, A383013, A384889, A384890.

Programs

  • Mathematica
    Table[Length[Select[Subsets[Range[n]],UnsameQ@@Length/@Split[#,#2==#1+1&]&]],{n,0,10}]
  • PARI
    lista(n)={my(o=(1-x^(n+1))/(1-x)*O(y^(n+2)),p=prod(i=1,n,1+o+x*y^(i+1)/(1-y),1/(1-y)));p=subst(serlaplace(p),x,1);Vec(p-1)} \\ Christian Sievers, Jun 18 2025

Extensions

a(21) and beyond from Christian Sievers, Jun 18 2025

A384177 Number of subsets of {1..n} with all distinct lengths of maximal anti-runs (increasing by more than 1).

Original entry on oeis.org

1, 2, 3, 5, 10, 19, 35, 62, 109, 197, 364, 677, 1251, 2288, 4143, 7443, 13318, 23837, 42809, 77216, 139751, 253293, 458800, 829237, 1494169, 2683316, 4804083, 8580293, 15301324, 27270061, 48607667, 86696300, 154758265, 276453311, 494050894, 882923051
Offset: 0

Views

Author

Gus Wiseman, Jun 16 2025

Keywords

Examples

			The subset {1,2,4,5,7,10} has maximal anti-runs ((1),(2,4),(5,7,10)), with lengths (1,2,3), so is counted under a(10).
The a(0) = 1 through a(5) = 19 subsets:
  {}  {}   {}   {}     {}       {}
      {1}  {1}  {1}    {1}      {1}
           {2}  {2}    {2}      {2}
                {3}    {3}      {3}
                {1,3}  {4}      {4}
                       {1,3}    {5}
                       {1,4}    {1,3}
                       {2,4}    {1,4}
                       {1,2,4}  {1,5}
                       {1,3,4}  {2,4}
                                {2,5}
                                {3,5}
                                {1,2,4}
                                {1,2,5}
                                {1,3,4}
                                {1,3,5}
                                {1,4,5}
                                {2,3,5}
                                {2,4,5}

Links

Crossrefs

For runs instead of anti-runs we have A384175, complement A384176.
These subsets are ranked by A384879.
For strict partitions instead of subsets we have A384880, see A384178, A384884, A384886.
For equal instead of distinct lengths we have A384889, for runs A243815.
A034839 counts subsets by number of maximal runs, for strict partitions A116674.
A098859 counts Wilf partitions (distinct multiplicities), complement A336866.
A384893 counts subsets by number of maximal anti-runs, for partitions A268193, A384905.
Cf. A000009, A010027, A044813, A047993, A106529, A123513, A242882, A325325, A328592, A329739, A351202, A384890.

Programs

  • Mathematica
    Table[Length[Select[Subsets[Range[n]],UnsameQ@@Length/@Split[#,#2!=#1+1&]&]],{n,0,10}]
  • PARI
    lista(n)={my(o=(1-x^(n+1))/(1-x)*O(y*y^n),p=prod(i=1,(n+1)\2,1+o+x*y^(2*i-1)/(1-y)^(i-1)));p=subst(serlaplace(p),x,1);Vec((p-y)/(1-y)^2)} \\ Christian Sievers, Jun 18 2025

Extensions

a(21) and beyond from Christian Sievers, Jun 18 2025

A384176 Number of subsets of {1..n} without all distinct lengths of maximal runs (increasing by 1).

Original entry on oeis.org

0, 0, 0, 1, 3, 8, 20, 51, 121, 276, 612, 1335, 2881, 6144, 12950, 27029, 55977, 115222, 236058, 481683, 979443
Offset: 0

Views

Author

Gus Wiseman, Jun 16 2025

Keywords

Examples

			The subset {1,3,4,8,9} has maximal runs ((1),(3,4),(8,9)), with lengths (1,2,2), so is counted under a(10).
The a(0) = 0 through a(6) = 20 subsets:
  .  .  .  {1,3}  {1,3}  {1,3}      {1,3}
                  {1,4}  {1,4}      {1,4}
                  {2,4}  {1,5}      {1,5}
                         {2,4}      {1,6}
                         {2,5}      {2,4}
                         {3,5}      {2,5}
                         {1,3,5}    {2,6}
                         {1,2,4,5}  {3,5}
                                    {3,6}
                                    {4,6}
                                    {1,3,5}
                                    {1,3,6}
                                    {1,4,6}
                                    {2,4,6}
                                    {1,2,4,5}
                                    {1,2,4,6}
                                    {1,2,5,6}
                                    {1,3,4,6}
                                    {1,3,5,6}
                                    {2,3,5,6}

Crossrefs

For equal instead of distinct lengths the complement is A243815.
These subsets are ranked by the non-members of A328592.
The complement is counted by A384175.
For strict partitions instead of subsets see A384178, A384884, A384886, A384880.
For permutations instead of subsets see A384891, A384892, A010027.
A034839 counts subsets by number of maximal runs, for strict partitions A116674.
A098859 counts Wilf partitions (distinct multiplicities), complement A336866.
A384893 counts subsets by number of maximal anti-runs, for partitions A268193, A384905.
Cf. A000009, A044813, A242882, A325325, A329739, A351202, A383013, A384177, A384889, A384890.

Programs

  • Mathematica
    Table[Length[Select[Subsets[Range[n]],!UnsameQ@@Length/@Split[#,#2==#1+1&]&]],{n,0,10}]

A384886 Number of strict integer partitions of n with all equal lengths of maximal runs (decreasing by 1).

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 4, 4, 4, 7, 7, 8, 11, 11, 14, 17, 19, 20, 27, 27, 35, 38, 45, 47, 60, 63, 75, 84, 97, 104, 127, 134, 155, 175, 196, 218, 251, 272, 307, 346, 384, 424, 480, 526, 586, 658, 719, 798, 890, 979, 1078, 1201, 1315, 1451, 1603, 1762, 1934, 2137
Offset: 0

Views

Author

Gus Wiseman, Jun 13 2025

Keywords

Examples

			The strict partition y = (7,6,5,3,2,1) has maximal runs ((7,6,5),(3,2,1)), with lengths (3,3), so y is counted under a(24).
The a(1) = 1 through a(14) = 14 partitions (A-E = 10-14):
  1  2  3   4   5   6    7   8   9    A     B    C     D    E
        21  31  32  42   43  53  54   64    65   75    76   86
                41  51   52  62  63   73    74   84    85   95
                    321  61  71  72   82    83   93    94   A4
                                 81   91    92   A2    A3   B3
                                 432  631   A1   B1    B2   C2
                                 531  4321  641  543   C1   D1
                                            731  642   742  752
                                                 741   751  842
                                                 831   841  851
                                                 5421  931  941
                                                            A31
                                                            5432
                                                            6521

Links

Crossrefs

For subsets instead of strict partitions we have A243815, distinct lengths A384175.
For distinct instead of equal lengths we have A384178, for anti-runs A384880.
This is the strict case of A384904, distinct lengths A384884.
A000041 counts integer partitions, strict A000009.
A047993 counts partitions with max part = length (A106529).
A098859 counts Wilf partitions (complement A336866), compositions A242882.
A239455 counts Look-and-Say or section-sum partitions, ranks A351294 or A381432.
Cf. A000217, A008284, A044813, A047966, A089259, A325324, A325325, A329739, A382857, A383013, A383708, A384176.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&SameQ@@Length/@Split[#,#2==#1-1&]&]],{n,0,15}]
  • PARI
    A_q(N) = {Vec(1+sum(k=1,floor(-1/2+sqrt(2+2*N)), sum(i=1,(N/(k*(k+1)/2))+1, q^(k*(k+1)*i^2/2)/prod(j=1,i, 1 - q^(j*k)))) + O('q^(N+1)))} \\ John Tyler Rascoe, Aug 21 2025

Formula

G.f.: 1 + Sum_{i,k>0} q^(k*(k+1)*i^2/2)/Product_{j=1..i} (1 - q^(j*k)). - John Tyler Rascoe, Aug 21 2025

A384884 Number of integer partitions of n with all distinct lengths of maximal gapless runs (decreasing by 0 or 1).

Original entry on oeis.org

1, 1, 2, 3, 4, 6, 9, 13, 18, 25, 35, 46, 60, 79, 104, 131, 170, 215, 271, 342, 431, 535, 670, 830, 1019, 1258, 1547, 1881, 2298, 2787, 3359, 4061, 4890, 5849, 7010, 8361, 9942, 11825, 14021, 16558, 19561, 23057, 27084, 31821, 37312, 43627, 50999, 59500, 69267
Offset: 0

Views

Author

Gus Wiseman, Jun 13 2025

Keywords

Examples

			The partition y = (6,6,4,3,3,2) has maximal gapless runs ((6,6),(4,3,3,2)), with lengths (2,4), so y is counted under a(24).
The a(1) = 1 through a(8) = 18 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (21)   (22)    (32)     (33)      (43)       (44)
             (111)  (211)   (221)    (222)     (322)      (332)
                    (1111)  (311)    (321)     (331)      (422)
                            (2111)   (411)     (421)      (431)
                            (11111)  (2211)    (511)      (521)
                                     (3111)    (2221)     (611)
                                     (21111)   (3211)     (2222)
                                     (111111)  (4111)     (3221)
                                               (22111)    (4211)
                                               (31111)    (5111)
                                               (211111)   (22211)
                                               (1111111)  (32111)
                                                          (41111)
                                                          (221111)
                                                          (311111)
                                                          (2111111)
                                                          (11111111)

Crossrefs

For subsets instead of strict partitions we have A384175.
The strict case is A384178, for anti-runs A384880.
For anti-runs we have A384885.
For equal instead of distinct lengths we have A384887.
A000041 counts integer partitions, strict A000009.
A007690 counts partitions with no singletons, complement A183558.
A034296 counts flat or gapless partitions, ranks A066311 or A073491.
A098859 counts Wilf partitions (distinct multiplicities), complement A336866.
A239455 counts Look-and-Say or section-sum partitions, ranks A351294 or A381432.
A355394 counts partitions without a neighborless part, singleton case A355393.
A356236 counts partitions with a neighborless part, singleton case A356235.
A356606 counts strict partitions without a neighborless part, complement A356607.
Cf. A008284, A044813, A047993, A242882, A287170, A325324, A325325, A356226, A356230, A356233, A356234, A384176, A384177, A384886.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],UnsameQ@@Length/@Split[#,#2>=#1-1&]&]],{n,0,15}]

A384178 Number of strict integer partitions of n with all distinct lengths of maximal runs (decreasing by 1).

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 2, 3, 3, 4, 5, 6, 6, 8, 8, 10, 11, 13, 13, 16, 15, 19, 19, 23, 22, 26, 28, 31, 35, 39, 37, 47, 51, 52, 60, 65, 67, 78, 85, 86, 99, 108, 110, 127, 136, 138, 159, 170, 171, 196, 209, 213, 240, 257, 260, 292, 306, 313, 350, 371, 369, 417, 441
Offset: 0

Views

Author

Gus Wiseman, Jun 12 2025

Keywords

Examples

			The strict partition y = (9,7,6,5,2,1) has maximal runs ((9),(7,6,5),(2,1)), with lengths (1,3,2), so y is counted under a(30).
The a(1) = 1 through a(14) = 8 strict partitions (A-E = 10-14):
  1  2  3   4  5   6    7    8    9    A     B     C     D     E
        21     32  321  43   431  54   532   65    543   76    653
                        421  521  432  541   542   651   643   743
                                  621  721   632   732   652   761
                                       4321  821   921   832   932
                                             5321  6321  A21   B21
                                                         5431  5432
                                                         7321  8321

Crossrefs

For subsets instead of strict partitions we have A384175, complement A384176.
For anti-runs instead of runs we have A384880.
This is the strict version of A384884.
For equal instead of distinct lengths we have A384886.
A000041 counts integer partitions, strict A000009.
A047993 counts partitions with max part = length.
A098859 counts Wilf partitions (complement A336866), compositions A242882.
A239455 counts Look-and-Say or section-sum partitions, ranks A351294 or A381432.
A351293 counts non-Look-and-Say or non-section-sum partitions, ranks A351295 or A381433.
Cf. A008284, A044813, A325324, A325325, A329739, A351202.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&UnsameQ@@Length/@Split[#,#1==#2+1&]&]],{n,0,30}]

A384889 Number of subsets of {1..n} with all equal lengths of maximal anti-runs (increasing by more than 1).

Original entry on oeis.org

1, 2, 4, 8, 14, 23, 37, 59, 93, 146, 230, 365, 584, 940, 1517, 2450, 3959, 6404, 10373, 16822, 27298, 44297, 71843, 116429, 188550, 305200, 493930, 799422, 1294108, 2095291, 3392736, 5493168, 8892148, 14390372, 23282110, 37660759, 60914308, 98528312, 159386110
Offset: 0

Views

Author

Gus Wiseman, Jun 18 2025

Keywords

Examples

			The subset {3,6,7,9,10,12} has maximal anti-runs ((3,6),(7,9),(10,12)), with lengths (2,2,2), so is counted under a(12).
The a(0) = 1 through a(4) = 14 subsets:
  {}  {}   {}     {}       {}
      {1}  {1}    {1}      {1}
           {2}    {2}      {2}
           {1,2}  {3}      {3}
                  {1,2}    {4}
                  {1,3}    {1,2}
                  {2,3}    {1,3}
                  {1,2,3}  {1,4}
                           {2,3}
                           {2,4}
                           {3,4}
                           {1,2,3}
                           {2,3,4}
                           {1,2,3,4}

Links

Crossrefs

For runs instead of anti-runs we have A243815, distinct A384175, complement A384176.
For distinct instead or equal lengths we have A384177, ranks A384879.
For partitions instead of subsets we have A384888.
A034296 counts flat or gapless partitions, ranks A066311 or A073491.
A034839 counts subsets by number of maximal runs, for strict partitions A116674.
A047966 counts uniform partitions (equal multiplicities), ranks A072774.
A384893 counts subsets by number of maximal anti-runs, for partitions A268193, A384905.
Cf. A010027, A044813, A047993, A356606, A384880, A384886, A384890.

Programs

  • Mathematica
    Table[Length[Select[Subsets[Range[n]],SameQ@@Length/@Split[#,#2!=#1+1&]&]],{n,0,10}]
  • PARI
    lista(n)=Vec(sum(i=1,(n+1)\2,1/(1-x^(2*i-1)/(1-x)^(i-1))-1,1-x+O(x*x^n))/(1-x)^2) \\ Christian Sievers, Jun 20 2025

Formula

G.f.: ( Sum_{i>=1} (1/(1-x^(2*i-1)/(1-x)^(i-1))-1) + 1-x ) / (1-x)^2. - Christian Sievers, Jun 21 2025

Extensions

a(21) and beyond from Christian Sievers, Jun 20 2025

A384887 Number of integer partitions of n with all equal lengths of maximal gapless runs (decreasing by 0 or 1).

Original entry on oeis.org

1, 1, 2, 3, 5, 6, 9, 10, 14, 18, 21, 26, 35, 39, 46, 58, 68, 79, 97, 111, 131, 155, 177, 206, 246, 278, 318, 373, 423, 483, 563, 632, 722, 827, 931, 1058, 1209, 1354, 1528, 1736, 1951, 2188, 2475, 2762, 3097, 3488, 3886, 4342, 4876, 5414, 6038, 6741, 7482
Offset: 0

Views

Author

Gus Wiseman, Jun 15 2025

Keywords

Examples

			The partition y = (6,5,5,5,3,3,2,1) has maximal gapless runs ((6,5,5,5),(3,3,2,1)), with lengths (4,4), so y is counted under a(30).
The a(1) = 1 through a(8) = 14 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (21)   (22)    (32)     (33)      (43)       (44)
             (111)  (31)    (41)     (42)      (52)       (53)
                    (211)   (221)    (51)      (61)       (62)
                    (1111)  (2111)   (222)     (322)      (71)
                            (11111)  (321)     (2221)     (332)
                                     (2211)    (3211)     (2222)
                                     (21111)   (22111)    (3221)
                                     (111111)  (211111)   (3311)
                                               (1111111)  (22211)
                                                          (32111)
                                                          (221111)
                                                          (2111111)
                                                          (11111111)

Crossrefs

The strict case is A384886, distinct A384178.
For distinct instead of equal lengths we have A384884.
For anti-runs instead of runs we have A384888, distinct A384885.
For subsets instead of strict partitions we have A243815.
Without counting decreases by 0 we get A384904.
A000041 counts integer partitions, strict A000009.
A007690 counts partitions with no singletons, complement A183558.
A034296 counts flat or gapless partitions, ranks A066311 or A073491.
A098859 counts Wilf partitions (distinct multiplicities), complement A336866.
A355394 counts partitions without a neighborless part, singleton case A355393.
A356236 counts partitions with a neighborless part, singleton case A356235.
A356606 counts strict partitions without a neighborless part, complement A356607.
Cf. A008284, A044813, A047993, A325324, A325325, A356226, A356230, A356233, A356234, A384175, A384177, A384880.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],SameQ@@Length/@Split[#,#2>=#1-1&]&]],{n,0,15}]

A384885 Number of integer partitions of n with all distinct lengths of maximal anti-runs (decreasing by more than 1).

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 4, 6, 8, 9, 13, 15, 18, 22, 28, 31, 38, 45, 53, 62, 74, 86, 105, 123, 146, 171, 208, 242, 290, 340, 399, 469, 552, 639, 747, 862, 999, 1150, 1326, 1514, 1736, 1979, 2256, 2560, 2909, 3283, 3721, 4191, 4726, 5311, 5973, 6691, 7510, 8396, 9395
Offset: 0

Views

Author

Gus Wiseman, Jun 13 2025

Keywords

Examples

			The partition y = (8,6,3,3,3,1) has maximal anti-runs ((8,6,3),(3),(3,1)), with lengths (3,1,2), so y is counted under a(24).
The partition z = (8,6,5,3,3,1) has maximal anti-runs ((8,6),(5,3),(3,1)), with lengths (2,2,2), so z is not counted under a(26).
The a(1) = 1 through a(9) = 9 partitions:
  (1)  (2)  (3)  (4)    (5)      (6)      (7)      (8)      (9)
                 (3,1)  (4,1)    (4,2)    (5,2)    (5,3)    (6,3)
                        (3,1,1)  (5,1)    (6,1)    (6,2)    (7,2)
                                 (4,1,1)  (3,3,1)  (7,1)    (8,1)
                                          (4,2,1)  (4,2,2)  (4,4,1)
                                          (5,1,1)  (4,3,1)  (5,2,2)
                                                   (5,2,1)  (5,3,1)
                                                   (6,1,1)  (6,2,1)
                                                            (7,1,1)

Crossrefs

For subsets instead of strict partitions we have A384177, for runs A384175.
The strict case is A384880.
For runs instead of anti-runs we have A384884, strict A384178.
For equal instead of distinct lengths we have A384888, for runs A384887.
A000041 counts integer partitions, strict A000009.
A007690 counts partitions with no singletons, complement A183558.
A034296 counts flat or gapless partitions, ranks A066311 or A073491.
A098859 counts Wilf partitions (distinct multiplicities), complement A336866.
A239455 counts Look-and-Say or section-sum partitions, ranks A351294 or A381432.
A355394 counts partitions without a neighborless part, singleton case A355393.
A356236 counts partitions with a neighborless part, singleton case A356235.
A356606 counts strict partitions without a neighborless part, complement A356607.
Cf. A008284, A047966, A242882, A287170, A325324, A325325, A329739, A356226, A356230, A356234, A384886.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],UnsameQ@@Length/@Split[#,#2<#1-1&]&]],{n,0,15}]

A384891 Number of permutations of {1..n} with all distinct lengths of maximal runs (increasing by 1).

Original entry on oeis.org

1, 1, 1, 3, 3, 5, 23, 25, 43, 63, 345, 365, 665, 949, 1513, 8175, 9003, 15929, 23399, 36949, 51043, 293715, 314697, 570353, 826817, 1318201, 1810393, 2766099, 14180139, 15600413, 27707879, 40501321, 63981955, 88599903, 134362569, 181491125, 923029217
Offset: 0

Views

Author

Gus Wiseman, Jun 19 2025

Keywords

Examples

			The permutation (1,2,6,7,8,9,3,4,5) has maximal runs ((1,2),(6,7,8,9),(3,4,5)), with lengths (2,4,3), so is counted under a(9).
The a(0) = 1 through a(7) = 25 permutations:
  ()  (1)  (12)  (123)  (1234)  (12345)  (123456)  (1234567)
                 (231)  (2341)  (23451)  (123564)  (1234675)
                 (312)  (4123)  (34512)  (123645)  (1234756)
                                (45123)  (124563)  (1245673)
                                (51234)  (126345)  (1273456)
                                         (145623)  (1456723)
                                         (156234)  (1672345)
                                         (231456)  (2314567)
                                         (234156)  (2345167)
                                         (234561)  (2345671)
                                         (312456)  (3124567)
                                         (345126)  (3456127)
                                         (345612)  (3456712)
                                         (412356)  (4567123)
                                         (451236)  (4567231)
                                         (456231)  (4567312)
                                         (456312)  (5123467)
                                         (561234)  (5612347)
                                         (562341)  (5671234)
                                         (564123)  (6712345)
                                         (612345)  (6723451)
                                         (634512)  (6751234)
                                         (645123)  (7123456)
                                                   (7345612)
                                                   (7561234)

Links

Crossrefs

Counting by number of maximal anti-runs gives A010027, for runs A123513.
For subsets instead of permutations we have A384175, complement A384176.
For partitions we have A384884 (anti-runs A384885), strict A384178 (anti-runs A384880).
For equal instead of distinct lengths we have A384892.
For anti-runs instead of runs we have A384907.
A000041 counts integer partitions, strict A000009.
A034839 counts subsets by number of maximal runs, for strict partitions A116674.
A098859 counts Wilf partitions (distinct multiplicities), complement A336866.
A356606 counts strict partitions without a neighborless part, complement A356607.
A384893 counts subsets by number of maximal anti-runs, for partitions A268193, A384905.
Cf. A000255, A044813, A072574, A242882, A287170, A325324, A325325, A328592, A329739, A351202, A384177, A384886.

Programs

  • Mathematica
    Table[Length[Select[Permutations[Range[n]],UnsameQ@@Length/@Split[#,#2==#1+1&]&]],{n,0,10}]
  • PARI
    lista(n)=my(b(n)=sum(i=0,n-1,(-1)^i*(n-i)!*binomial(n-1,i)), d=floor(sqrt(2*n)), p=prod(i=1,n,1+x*y^i,1+O(y*y^n)*((1-x^(n+1))/(1-x))+O(x*x^d))); Vec(1+sum(i=1,d,i!*b(i)*polcoef(p,i))) \\ Christian Sievers, Jun 22 2025

Formula

a(n) = Sum_{k=1..n} ( T(n,k) * A000255(k-1) ) for n>=1, where T(n,k) is the number of compositions of n into k distinct parts (cf. A072574). - Christian Sievers, Jun 22 2025

Extensions

a(11) and beyond from Christian Sievers, Jun 22 2025
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