A384175 -id:A384175 - cp's OEIS frontend

A209859 Rewrite the binary expansion of n from the most significant end, 1 -> 1, 0+1 (one or more zeros followed by one) -> 0, drop the trailing zeros of the original n.

0, 1, 1, 3, 1, 2, 3, 7, 1, 2, 2, 5, 3, 6, 7, 15, 1, 2, 2, 5, 2, 4, 5, 11, 3, 6, 6, 13, 7, 14, 15, 31, 1, 2, 2, 5, 2, 4, 5, 11, 2, 4, 4, 9, 5, 10, 11, 23, 3, 6, 6, 13, 6, 12, 13, 27, 7, 14, 14, 29, 15, 30, 31, 63, 1, 2, 2, 5, 2, 4, 5, 11, 2, 4, 4, 9, 5, 10, 11, 23, 2, 4, 4, 9, 4, 8, 9, 19, 5, 10, 10, 21, 11, 22, 23, 47, 3, 6, 6, 13, 6, 12, 13, 27, 6, 12, 12, 25, 13

Offset: 0

Antti Karttunen, Mar 24 2012

This is the number k such that the k-th composition in standard order is the reversed sequence of lengths of the maximal anti-runs of the binary indices of n. Here, the binary indices of n are row n of A048793, and the k-th composition in standard order is row k of A066099. For example, the binary indices of 100 are {3,6,7}, with maximal anti-runs ((3,6),(7)), with reversed lengths (1,2), which is the 6th composition in standard order, so a(100) = 6. - Gus Wiseman, Jul 27 2025

			102 in binary is 1100110, we rewrite it from the left so that first two 1's stay same ("11"), then "001" is rewritten to "0", the last 1 to "1", and we ignore the last 0, thus getting 1101, which is binary expansion of 13, thus a(102) = 13.

Gus Wiseman, Statistics, classes, and transformations of standard compositions

This is an "inverse" of A071162, i.e. a(A071162(n)) = n for all n. Bisection: A209639. Used to construct permutation A209862.
Removing duplicates appears to give A358654.
Sorted positions of firsts appearances appear to be A247648+1.
A245563 lists run-lengths of binary indices (ranks A246029), reverse A245562.
A384175 counts subsets with all distinct lengths of maximal runs, complement A384176.
Cf. A000120, A023758, A029931, A044813, A048793, A052499, A069010, A348366, A384878, A384879, A384893.
Cf. A384877, A385816, A385887, A385889.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
stcinv[q_]:=Total[2^(Accumulate[Reverse[q]])]/2;
Table[stcinv[Reverse[Length/@Split[bpe[n],#2!=#1+1&]]],{n,0,100}] (* Gus Wiseman, Jul 25 2025 *)

import re
def a(n): return int(re.sub("0+1", "0", bin(n)[2:].rstrip("0")), 2) if n else 0
print([a(n) for n in range(109)])  # Michael S. Branicky, Jul 25 2025

(define (A209859 n) (let loop ((n n) (s 0) (i (A053644 n))) (cond ((zero? n) s) ((> i n) (if (> (/ i 2) n) (loop n s (/ i 2)) (loop (- n (/ i 2)) (* 2 s) (/ i 4)))) (else (loop (- n i) (+ (* 2 s) 1) (/ i 2))))))

a(n) = a(A000265(n)).

A385889 The number k such that the k-th composition in standard order is the sequence of lengths of maximal runs of binary indices of n.

Original entry on oeis.org

0, 1, 1, 2, 1, 3, 2, 4, 1, 3, 3, 5, 2, 6, 4, 8, 1, 3, 3, 5, 3, 7, 5, 9, 2, 6, 6, 10, 4, 12, 8, 16, 1, 3, 3, 5, 3, 7, 5, 9, 3, 7, 7, 11, 5, 13, 9, 17, 2, 6, 6, 10, 6, 14, 10, 18, 4, 12, 12, 20, 8, 24, 16, 32, 1, 3, 3, 5, 3, 7, 5, 9, 3, 7, 7, 11, 5, 13, 9, 17, 3

Offset: 0

Gus Wiseman, Jul 16 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 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 binary indices of 27 are {1,2,4,5}, with maximal runs ((1,2),(4,5)), with lengths (2,2), which is the 10th composition in standard order, so a(27) = 10.
The binary indices of 100 are {3,6,7}, with maximal runs ((3),(6,7)), with lengths (1,2), which is the 6th composition in standard order, so a(100) = 6.

Gus Wiseman, Statistics, classes, and transformations of standard compositions

Sorted positions of firsts appearances appear to be A247648+1.
After removing duplicates we get A385818.
The reverse version is A385887.
A245563 lists run lengths of binary indices (ranks A246029), reverse A245562.
A384877 lists anti-run lengths of binary indices (ranks A385816), reverse A209859.
Cf. A000120, A023758, A029931, A044813, A048793, A069010, A348366, A384175, A384878, A385817.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
stcinv[q_]:=Total[2^(Accumulate[Reverse[q]])]/2;
Table[stcinv[Length/@Split[bpe[n],#2==#1+1&]],{n,0,100}]

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

Gus Wiseman, Jun 19 2025

nonn

			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)

Christian Sievers, Table of n, a(n) for n = 0..5000

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.

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

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

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

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

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

Gus Wiseman, Jun 19 2025

nonn

			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)

Christian Sievers, Table of n, a(n) for n = 0..450

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.

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

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

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

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

Gus Wiseman, Jun 20 2025

nonn

			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)

John Tyler Rascoe, Table of n, a(n) for n = 0..100

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.

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

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

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

Gus Wiseman, Jul 14 2025

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.

			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

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.

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

A385818 The number k such that the k-th composition in standard order lists the maximal run lengths of each nonnegative integer's binary indices, with duplicates removed.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 8, 7, 9, 10, 12, 16, 11, 13, 17, 14, 18, 20, 24, 32, 15, 19, 21, 25, 33, 22, 26, 34, 28, 36, 40, 48, 64, 23, 27, 35, 29, 37, 41, 49, 65, 30, 38, 42, 50, 66, 44, 52, 68, 56, 72, 80, 96, 128, 31, 39, 43, 51, 67, 45, 53, 69, 57, 73, 81, 97

Offset: 0

Gus Wiseman, Jul 18 2025

A permutation of the nonnegative integers.
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 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 binary indices of 53 are {1,3,5,6}, with maximal runs ((1),(3),(5,6)) with lengths (1,1,2), which is the 14th composition in standard order, so A385889(53) = 14, and after removing duplicate rows a(16) = 14.

John Tyler Rascoe, Table of n, a(n) for n = 0..8192

For anti-runs instead of runs we appear to have A348366.
See also A385816 (standard compositions of rows of A384877), reverse A209859.
The compositions themselves are listed by A385817.
Before removing duplicates we had A385889.
A245563 lists run lengths of binary indices (ranks A246029), rev A245562, strict A328592.
A384175 counts subsets with all distinct lengths of maximal runs, complement A384176.
Cf. A000120, A029931, A048793, A052499, A069010, A200648, A200649, A200650, A384890, A385886, A385887.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
stcinv[q_]:=Total[2^(Accumulate[Reverse[q]])]/2;
stcinv/@DeleteDuplicates[Table[Length/@Split[bpe[n],#2==#1+1&],{n,0,100}]]

A202023 Triangle T(n,k), read by rows, given by (1, 0, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 3, 0, 0, 1, 6, 1, 0, 0, 1, 10, 5, 0, 0, 0, 1, 15, 15, 1, 0, 0, 0, 1, 21, 35, 7, 0, 0, 0, 0, 1, 28, 70, 28, 1, 0, 0, 0, 0, 1, 36, 126, 84, 9, 0, 0, 0, 0, 0, 1, 45, 210, 210, 45, 1, 0, 0, 0, 0, 0

Offset: 0

Philippe Deléham, Dec 10 2011

Riordan array (1/(1-x), x^2/(1-x)^2).
A skewed version of triangular array A085478.
Mirror image of triangle in A098158.
Sum_{k, 0<=k<=n} T(n,k)*x^k = A138229(n), A006495(n), A138230(n),A087455(n), A146559(n), A000012(n), A011782(n), A001333(n),A026150(n), A046717(n), A084057(n), A002533(n), A083098(n),A084058(n), A003665(n), A002535(n), A133294(n), A090042(n),A125816(n), A133343(n), A133345(n), A120612(n), A133356(n), A125818(n) for x = -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18 respectively.
Sum_{k, 0<=k<=n} T(n,k)*x^(n-k) = A009116(n), A000007(n), A011782(n), A006012(n), A083881(n), A081335(n), A090139(n), A145301(n), A145302(n), A145303(n), A143079(n) for x = -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 respectively.
From Gus Wiseman, Jul 08 2025: (Start)
After the first row this is also the number of subsets of {1..n-1} with k maximal runs (sequences of consecutive elements increasing by 1) for k = 0..n. For example, row n = 5 counts the following subsets:
  {}  {1}        {1,3}    .  .  .
      {2}        {1,4}
      {3}        {2,4}
      {4}        {1,2,4}
      {1,2}      {1,3,4}
      {2,3}
      {3,4}
      {1,2,3}
      {2,3,4}
      {1,2,3,4}
Requiring n-1 gives A202064.
For anti-runs instead of runs we have A384893.
(End)

			Triangle begins :
1
1, 0
1, 1, 0
1, 3, 0, 0
1, 6, 1, 0, 0
1, 10, 5, 0, 0, 0
1, 15, 15, 1, 0, 0, 0
1, 21, 35, 7, 0, 0, 0, 0
1, 28, 70, 28, 1, 0, 0, 0, 0

Column k = 1 is A000217.
Column k = 2 is A000332.
Row sums are A011782 (or A000079 shifted right).
Removing all zeros gives A034839 (requiring n-1 A034867).
Last nonzero term in each row appears to be A093178, requiring n-1 A124625.
Reversing rows gives A098158, without zeros A109446.
Without the k = 0 column we get A210039.
Row maxima appear to be A214282.
A116674 counts strict partitions by number of maximal runs, for anti-runs A384905.
A268193 counts integer partitions by number of maximal runs, for anti-runs A384881.
Cf. A000045, A000071, A007318, A010027, A053538, A084938, A202064, A208342, A210034, A384175, A384893.

Table[Length[Select[Subsets[Range[n-1]],Length[Split[#,#2==#1+1&]]==k&]],{n,0,10},{k,0,n}] (* Gus Wiseman, Jul 08 2025 *)

T(n,k) = binomial(n,2k).
G.f.: (1-x)/((1-x)^2-y*x^2).
T(n,k)= Sum_{j, j>=0} T(n-1-j,k-1)*j with T(n,0)=1 and T(n,k)= 0 if k<0 or if nT(n,k) = 2*T(n-1,k) + T(n-2,k-1) - T(n-2,k) for n>1, T(0,0) = T(1,0) = 1, T(1,1) = 0, T(n,k) = 0 if k>n or if k<0. - Philippe Deléham, Nov 10 2013

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

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}]

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

Gus Wiseman, Jun 20 2025

nonn

			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

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.

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

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A209859 Rewrite the binary expansion of n from the most significant end, 1 -> 1, 0+1 (one or more zeros followed by one) -> 0, drop the trailing zeros of the original n.

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A385889 The number k such that the k-th composition in standard order is the sequence of lengths of maximal runs of binary indices of n.

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

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

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

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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.

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A385818 The number k such that the k-th composition in standard order lists the maximal run lengths of each nonnegative integer's binary indices, with duplicates removed.

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A202023 Triangle T(n,k), read by rows, given by (1, 0, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

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