A384176 -id:A384176 - cp's OEIS frontend

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

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

Gus Wiseman, Jun 15 2025

nonn

			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)

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.

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

A385572 Number of subsets of {1..n} with the same number of maximal runs (increasing by 1) as maximal anti-runs (increasing by more than 1).

Original entry on oeis.org

1, 2, 3, 4, 7, 12, 19, 34, 63, 112, 207, 394, 739, 1398, 2687, 5152, 9891, 19128, 37039, 71754, 139459, 271522, 528999, 1032308, 2017291, 3945186, 7723203, 15134440, 29679407, 58245068, 114389683, 224796210, 442021743, 869658304, 1711914351, 3371515306

Offset: 0

Gus Wiseman, Jul 04 2025

nonn

Also the number of subsets of {1..n} with the same number of adjacent elements increasing by 1 as adjacent elements increasing by more than 1.

			The set {2,3,5,6,8} has maximal runs ((2,3),(5,6),(8)) and maximal anti-runs ((2),(3,5),(6,8)) so is counted under a(8).
The a(0) = 1 through a(6) = 19 subsets:
  {}  {}   {}   {}   {}       {}       {}
      {1}  {1}  {1}  {1}      {1}      {1}
           {2}  {2}  {2}      {2}      {2}
                {3}  {3}      {3}      {3}
                     {4}      {4}      {4}
                     {1,2,4}  {5}      {5}
                     {1,3,4}  {1,2,4}  {6}
                              {1,2,5}  {1,2,4}
                              {1,3,4}  {1,2,5}
                              {1,4,5}  {1,2,6}
                              {2,3,5}  {1,3,4}
                              {2,4,5}  {1,4,5}
                                       {1,5,6}
                                       {2,3,5}
                                       {2,3,6}
                                       {2,4,5}
                                       {2,5,6}
                                       {3,4,6}
                                       {3,5,6}

Christian Sievers, Table of n, a(n) for n = 0..3328

The LHS is counted by A034839 (for partitions A384881, strict A116674), rank statistic A069010.
The case containing n + 1 is A217615.
The RHS is counted by A384893 or A210034 (for partitions A268193, strict A384905), rank statistic A384890.
Subsets of this type are ranked by A385575.
A384175 counts subsets with all distinct lengths of maximal runs, complement A384176.
A384877 gives lengths of maximal anti-runs in binary indices, firsts A384878.
Cf. A000071, A000079, A001629, A010027, A053538, A384177, A384879, A384889.

a:= proc(n) option remember; `if`(n<5, [1, 2, 3, 4, 7][n+1], ((3*n-4)*a(n-1)-
      (3*n-5)*a(n-2)+(5*n-12)*a(n-3)-2*(4*n-11)*a(n-4)+4*(n-3)*a(n-5))/(n-1))
    end:
seq(a(n), n=0..35);  # Alois P. Heinz, Jul 06 2025

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

a(n)=polcoef([1,1,1]*[x,0,0;x,x^2,1;0,x,x]^n*[1,0,0]~,n) \\ Christian Sievers, Jul 06 2025

Let M be the matrix [1,0,0; 1,x,1/x; 0,1,1]. Then a(n) is the sum of the constant terms of the entries in the left column of M^n. - Christian Sievers, Jul 06 2025

a(21) and beyond from Christian Sievers, Jul 06 2025

A385575 Numbers whose binary indices have the same number of adjacent parts differing by 1 as adjacent parts differing by more than 1.

Original entry on oeis.org

1, 2, 4, 8, 11, 13, 16, 19, 22, 25, 26, 32, 35, 38, 44, 49, 50, 52, 64, 67, 70, 76, 87, 88, 91, 93, 97, 98, 100, 104, 107, 109, 117, 128, 131, 134, 140, 151, 152, 155, 157, 167, 174, 176, 179, 182, 185, 186, 193, 194, 196, 200, 203, 205, 208, 211, 214, 217

Offset: 1

Gus Wiseman, Jul 04 2025

nonn

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.

			The terms together with their binary expansions and binary indices begin:
    1:       1 ~ {1}
    2:      10 ~ {2}
    4:     100 ~ {3}
    8:    1000 ~ {4}
   11:    1011 ~ {1,2,4}
   13:    1101 ~ {1,3,4}
   16:   10000 ~ {5}
   19:   10011 ~ {1,2,5}
   22:   10110 ~ {2,3,5}
   25:   11001 ~ {1,4,5}
   26:   11010 ~ {2,4,5}
   32:  100000 ~ {6}
   35:  100011 ~ {1,2,6}
   38:  100110 ~ {2,3,6}
   44:  101100 ~ {3,4,6}
   49:  110001 ~ {1,5,6}
   50:  110010 ~ {2,5,6}
   52:  110100 ~ {3,5,6}
   64: 1000000 ~ {7}
   67: 1000011 ~ {1,2,7}
   70: 1000110 ~ {2,3,7}
   76: 1001100 ~ {3,4,7}
   87: 1010111 ~ {1,2,3,5,7}
   88: 1011000 ~ {4,5,7}
   91: 1011011 ~ {1,2,4,5,7}
   93: 1011101 ~ {1,3,4,5,7}
   97: 1100001 ~ {1,6,7}
   98: 1100010 ~ {2,6,7}
  100: 1100100 ~ {3,6,7}

The LHS rank statistic is A069010, counted by A034839 (for partitions A384881, A116674).
The RHS rank statistic is A384890, counted by A384893 (for partitions A268193, A384905).
Subsets of this type are counted by A385572, with n A217615.
A384175 counts subsets with all distinct lengths of maximal runs, complement A384176.
A384877 gives lengths of maximal anti-runs in binary indices, firsts A384878.
Cf. A000079, A010027, A053538, A210034, A384177, A384879, A384889.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Select[Range[100],Length[Split[bpe[#],#2==#1+1&]]==Length[Split[bpe[#],#2!=#1+1&]]&]

is_ok(n)=hammingweight(n)==2*hammingweight(bitand(n,n>>1))+1 \\ Christian Sievers, Jul 18 2025

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

Gus Wiseman, Jun 25 2025

			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}

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.

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

Gus Wiseman, Jul 18 2025

nonn

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.

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.

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[[#]]]&]

cp's OEIS Frontend

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

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A385572 Number of subsets of {1..n} with the same number of maximal runs (increasing by 1) as maximal anti-runs (increasing by more than 1).

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A385575 Numbers whose binary indices have the same number of adjacent parts differing by 1 as adjacent parts differing by more than 1.

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

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

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