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.

Previous Showing 11-17 of 17 results.

A384892 Number of permutations of {1..n} with all equal lengths of maximal runs (increasing by 1).

Original entry on oeis.org

1, 1, 2, 4, 13, 54, 314, 2120, 16700, 148333, 1468512, 16019532, 190899736, 2467007774, 34361896102, 513137616840, 8178130784179, 138547156531410, 2486151753462260, 47106033220679060, 939765362754015750, 19690321886243848784, 432292066866187743954
Offset: 0

Views

Author

Gus Wiseman, Jun 19 2025

Keywords

Examples

			The permutation (1,2,5,6,3,4,7,8) has maximal runs ((1,2),(5,6),(3,4),(7,8)), with lengths (2,2,2,2), so is counted under a(8).
The a(0) = 1 through a(4) = 13 permutations:
  ()  (1)  (12)  (123)  (1234)
           (21)  (132)  (1324)
                 (213)  (1432)
                 (321)  (2143)
                        (2413)
                        (2431)
                        (3142)
                        (3214)
                        (3241)
                        (3412)
                        (4132)
                        (4213)
                        (4321)

Links

Crossrefs

For subsets instead of permutations we have A243815, for anti-runs A384889.
For strict partitions and distinct lengths we have A384178, anti-runs A384880.
For integer partitions and distinct lengths we have A384884, anti-runs A384885.
For distinct lengths we have A384891, for anti-runs A384907.
For partitions we have A384904, strict A384886.
A010027 counts permutations by maximal anti-runs, for runs A123513.
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. A000255, A044813, A047993, A242882, A325325, A329739, A351202, A384175, A384176, A384177.

Programs

  • Mathematica
    Table[Length[Select[Permutations[Range[n]],SameQ@@Length/@Split[#,#2==#1+1&]&]],{n,0,10}]
  • PARI
    a(n)=if(n,sumdiv(n,d,sum(i=0,d-1,(-1)^i*(d-i)!*binomial(d-1,i))),1) \\ Christian Sievers, Jun 22 2025

Formula

a(n) = Sum_{d|n} A000255(d-1). - Christian Sievers, Jun 22 2025

Extensions

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

A384904 Number of integer partitions of n with all equal lengths of maximal runs of consecutive parts decreasing by 1 but not by 0.

Original entry on oeis.org

1, 1, 2, 3, 4, 5, 9, 9, 14, 17, 23, 25, 40, 41, 59, 68, 92, 99, 140, 151, 204, 229, 296, 328, 433, 476, 606, 685, 858, 955, 1203, 1336, 1654, 1858, 2266, 2537, 3102, 3453, 4169, 4680, 5611, 6262, 7495, 8358, 9927, 11105, 13096, 14613, 17227, 19179, 22459
Offset: 0

Views

Author

Gus Wiseman, Jun 20 2025

Keywords

Examples

			The partition (6,5,5,4,2,1) has maximal runs ((6,5),(5,4),(2,1)), with lengths (2,2,2), so is counted under a(23).
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)
                    (1111)  (311)    (51)      (61)       (62)
                            (11111)  (222)     (331)      (71)
                                     (321)     (511)      (422)
                                     (411)     (4111)     (611)
                                     (3111)    (31111)    (2222)
                                     (111111)  (1111111)  (3221)
                                                          (3311)
                                                          (5111)
                                                          (41111)
                                                          (311111)
                                                          (11111111)

Links

Crossrefs

For subsets instead of strict partitions we have A243815, distinct lengths A384175.
For distinct instead of equal lengths we have A384882, counting gaps of 0 A384884.
The strict case is A384886.
Counting gaps of 0 gives A384887.
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, A047966, A089259, A325325, A382857, A383013, A383708, A384178, A384880.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],SameQ@@Length/@Split[#,#2==#1-1&]&]],{n,0,30}]
  • 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*i*(2+i*(k-1)))/2)/(1-q^(k*i))*prod(j=1,i-1, 1 + q^(2*k*j)/(1 - q^(k*j))))) + O('q^(N+1)))} \\ John Tyler Rascoe, Aug 20 2025

Formula

G.f.: 1 + Sum_{i,k>0} q^((i*k*(2 + i*(k-1)))/2) * Product_{j=1..i-1} ( 1 + q^(2*k*j)/(1 - q^(k*j)) ) / (1 - q^(i*k)). - John Tyler Rascoe, Aug 20 2025

A385817 Irregular triangle read by rows listing the lengths of maximal runs (sequences of consecutive elements increasing by 1) of binary indices, duplicate rows removed.

Original entry on oeis.org

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

Views

Author

Gus Wiseman, Jul 14 2025

Keywords

Comments

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.
This is the triangle A245563, except all duplicates after the first instance of each composition are removed. It lists all compositions in order of their first appearance as a row of A245563.

Examples

			The binary indices of 53 are {1,3,5,6}, with maximal runs ((1),(3),(5,6)), with lengths (1,1,2). After removing duplicates, this is our row 16.
Triangle begins:
   0: .
   1: 1
   2: 2
   3: 1 1
   4: 3
   5: 2 1
   6: 1 2
   7: 4
   8: 1 1 1
   9: 3 1
  10: 2 2
  11: 1 3
  12: 5
  13: 2 1 1
  14: 1 2 1
  15: 4 1
  16: 1 1 2
  17: 3 2
  18: 2 3
  19: 1 4
  20: 6
  21: 1 1 1 1

Crossrefs

In the following references, "before" is short for "before removing duplicate rows".
Positions of singleton rows appear to be A000071 = A000045-1, before A023758.
Positions of firsts appearances appear to be A001629.
Positions of rows of the form (1,1,...) appear to be A055588 = A001906+1.
First term of each row appears to be A083368.
Row sums appear to be A200648, before A000120.
Row lengths after the first row appear to be A200650+1, before A069010 = A037800+1.
Before the removals we had A245563 (except first term), see A245562, A246029, A328592.
For anti-run ranks we have A385816, before A348366, firsts A052499.
Standard composition numbers of rows are A385818, before A385889.
For anti-runs we have A385886, before A384877, firsts A384878.
Cf. A044813, A048793, A164707, A200649, A242882, A243815, A384175, A384890.

Programs

  • Mathematica
    DeleteDuplicates[Table[Length/@Split[Join@@Position[Reverse[IntegerDigits[n,2]],1],#2==#1+1&],{n,0,100}]]

A384888 Number of integer partitions of n with all equal lengths of maximal anti-runs (decreasing by more than 1).

Original entry on oeis.org

1, 1, 2, 3, 5, 6, 9, 10, 13, 17, 20, 24, 32, 36, 44, 55, 64, 75, 92, 105, 125, 147, 169, 195, 231, 263, 303, 351, 401, 458, 532, 600, 686, 784, 889, 1010, 1152, 1296, 1468, 1662, 1875, 2108, 2384, 2669, 3001, 3373, 3775, 4222, 4734, 5278, 5896, 6576, 7322
Offset: 0

Views

Author

Gus Wiseman, Jun 15 2025

Keywords

Examples

			The partition y = (10,6,6,4,3,1) has maximal anti-runs ((10,6),(6,4),(3,1)), with lengths (2,2,2), so y is counted under a(30).
The a(1) = 1 through a(8) = 13 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)   (22211)
                                               (1111111)  (32111)
                                                          (221111)
                                                          (2111111)
                                                          (11111111)

Crossrefs

The strict case is new, distinct A384880.
For distinct instead of equal lengths we have A384885.
For runs instead of anti-runs we have A384887, distinct A384884.
For subsets instead of strict partitions we have A384889, distinct A384177, runs A243815.
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, A325325, A356226, A384175, A384176, A384178, A384886.

Programs

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

A384882 Number of integer partitions of n with all distinct lengths of maximal runs of consecutive parts decreasing by 1 but not by 0.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 2, 5, 4, 5, 6, 9, 7, 12, 12, 11, 16, 18, 17, 25, 25, 23, 33, 35, 36, 42, 52, 45, 58, 64, 60, 77, 91, 79, 109, 108, 105, 129, 149, 134, 170, 179, 177, 213, 236, 208, 275, 281, 282, 323, 359, 330, 410, 433, 440, 474, 541, 508, 614, 631, 635
Offset: 0

Views

Author

Gus Wiseman, Jun 20 2025

Keywords

Examples

			The partition (6,5,5,5,3,2) has maximal runs ((6,5),(5),(5),(3,2)), with lengths (2,1,1,2), so is not counted under a(26).
The partition (6,5,5,5,4,3,2) has maximal runs ((6,5),(5),(5,4,3,2)), with lengths (2,1,4), so is counted under a(30).
The a(1) = 1 through a(13) = 12 partitions:
  1  2  3   4    5    6    7     8    9     A     B      C      D
        21  211  32   321  43    332  54    433   65     543    76
                 221       322   431  432   532   443    651    544
                           421   521  621   541   542    732    643
                           3211       3321  721   632    921    652
                                            4321  821    6321   832
                                                  4322   43221  A21
                                                  5321          4432
                                                  43211         5431
                                                                7321
                                                                43321
                                                                432211

Crossrefs

For subsets instead of strict partitions we have A384175, equal lengths A243815.
The strict case is A384178, for anti-runs A384880.
Counting gaps of 0 gives A384884, equal A384887.
For equal instead of distinct lengths we have A384904, strict case A384886.
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. A008284, A047966, A089259, A325325, A382857, A383013, A383708.

Programs

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

A385214 Number of subsets of {1..n} without all equal lengths of maximal runs of consecutive elements increasing by 1.

Original entry on oeis.org

0, 0, 0, 0, 2, 8, 25, 66, 159, 361, 791, 1688, 3539, 7328, 15040, 30669, 62246, 125896, 253975, 511357, 1028052
Offset: 0

Views

Author

Gus Wiseman, Jun 25 2025

Keywords

Examples

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

Crossrefs

These subsets are ranked by A164708, complement A164707
The complement is counted by A243815.
For distinct instead of equal lengths we have A384176, complement A384175.
For anti-runs instead of runs we have complement of A384889, for partitions A384888.
For permutations instead of subsets we have complement of A384892, distinct A384891.
For partitions instead of subsets we have complement of A384904, strict A384886.
A034839 counts subsets by number of maximal runs, for strict partitions A116674.
A049988 counts partitions with equal run-lengths, distinct A325325.
A329738 counts compositions with equal run-lengths, distinct A329739.
A384177 counts subsets with all distinct lengths of maximal anti-runs, ranks A384879.
A384887 counts partitions with equal lengths of gapless runs, distinct A384884.
A384893 counts subsets by number of maximal anti-runs, for partitions A268193, A384905.
Cf. A010027, A044813, A164710, A383013, A384885.

Programs

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

A385890 Positions of first appearances in A245563 = run lengths of binary indices.

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 14, 16, 22, 24, 28, 30, 32, 44, 46, 48, 54, 56, 60, 62, 64, 86, 88, 92, 94, 96, 108, 110, 112, 118, 120, 124, 126, 128, 172, 174, 176, 182, 184, 188, 190, 192, 214, 216, 220, 222, 224, 236, 238, 240, 246, 248, 252, 254, 256, 342, 344, 348
Offset: 1

Views

Author

Gus Wiseman, Jul 18 2025

Keywords

Comments

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.

Crossrefs

These are positions of firsts appearances in A245563, ranks A385889, reverse A245562.
A384175 counts subsets with all distinct lengths of maximal runs, complement A384176.
A384877 lists anti-run lengths of binary indices, ranks A385816.
Cf. A000120, A023758, A048793, A069010, A200649, A243815, A246029, A328592, A384890.

Programs

  • Mathematica
    bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
q=Table[Length/@Split[bpe[n],#2==#1+1&],{n,0,1000}];
Select[Range[Length[q]-1],!MemberQ[Take[q,#-1],q[[#]]]&]
Previous Showing 11-17 of 17 results.

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