A351203 -id:A351203 - cp's OEIS frontend

A351592 Number of Look-and-Say partitions (A239455) of n without distinct multiplicities, i.e., those that are not Wilf partitions (A098859).

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 3, 1, 0, 5, 2, 8, 9, 8, 6, 21, 14, 20, 26, 31, 24, 53, 35, 60, 68, 78, 76, 140, 115, 163, 183, 232, 218, 343, 301, 433, 432, 565, 542, 774, 728, 958, 977, 1251, 1220, 1612, 1561, 2053, 2090, 2618, 2609, 3326, 3378

Offset: 0

Gus Wiseman, Feb 16 2022

nonn

A partition is Look-and-Say iff it has a permutation with all distinct run-lengths. For example, the partition y = (2,2,2,1,1,1) has the permutation (2,2,1,1,1,2), with run-lengths (2,3,1), which are distinct, so y is counted under A239455(9).
A partition is Wilf iff it has distinct multiplicities of parts. For example, (2,2,2,1,1,1) has multiplicities (3,3), so is not counted under A098859(9).
The Heinz numbers of these partitions are given by A351294 \ A130091.
Is a(17) = 0 the last zero of the sequence?

			The a(9) = 1 through a(18) = 5 partitions are (empty columns not shown):
  n=9:      n=12:       n=15:         n=16:       n=18:
  --------------------------------------------------------------
  (222111)  (333111)    (333222)      (33331111)  (444222)
            (22221111)  (444111)                  (555111)
                        (2222211111)              (3322221111)
                                                  (32222211111)
                                                  (222222111111)

Wilf partitions are counted by A098859, ranked by A130091.
Look-and-Say partitions are counted by A239455, ranked by A351294.
Non-Wilf partitions are counted by A336866, ranked by A130092.
Non-Look-and-Say partitions are counted by A351293, ranked by A351295.
A000569 = number of graphical partitions, complement A339617.
A032020 = number of binary expansions with all distinct run-lengths.
A044813 = numbers whose binary expansion has all distinct run-lengths.
A225485/A325280 = frequency depth, ranked by A182850/A323014.
A329738 = compositions with all equal run-lengths.
A329739 = compositions with all distinct run-lengths
A351013 = compositions with all distinct runs.
A351017 = binary words with all distinct run-lengths, for all runs A351016.
A351292 = patterns with all distinct run-lengths, for all runs A351200.
Cf. A000041, A008284, A018783, A047966, A181819, A182857, A238130, A305563, A319149, A351203, A351204, A351290.

Table[Length[Select[IntegerPartitions[n], !UnsameQ@@Length/@Split[#]&&Select[Permutations[#], UnsameQ@@Length/@Split[#]&]!={}&]],{n,0,15}]

a(n) = A239455(n) - A098859(n). Here we assume A239455(0) = 1.

More terms from Jinyuan Wang, Feb 14 2025

A165933 Least integer, k, whose value is n in A165413.

Original entry on oeis.org

1, 4, 35, 536, 16775, 1060976, 135007759, 34460631520, 17617985239071, 18027600169142208, 36907002795598798911, 151143401509104346210176, 1238053384151947477501575295, 20283338091738780737237428502272, 664629209970464486086782992577855743

Offset: 1

Robert G. Wilson v, Sep 30 2009

An alternative name: The smallest number whose binary expansion has exactly n distinct run-lengths. - Gus Wiseman, Feb 21 2022

Term a(n) has one 1, followed by n 0's, then two 1's, (n-1) 0's, ..., up to n runs; see Python program. - Michael S. Branicky, Feb 22 2022

			a(1) in binary is 1, a(2) in binary is 100, a(3) in binary is 100011, a(4) in binary is 1000011000, etc.
From _Gus Wiseman_, Feb 21 2022: (Start)
The terms and their binary expansions begin:
  n              a(n)
  1:               1 =                                             1
  2:               4 =                                           100
  3:              35 =                                        100011
  4:             536 =                                    1000011000
  5:           16775 =                               100000110000111
  6:         1060976 =                         100000011000001110000
  7:       135007759 =                  1000000011000000111000001111
  8:     34460631520 =          100000000110000000111000000111100000
  9:  17617985239071 = 100000000011000000001110000000111100000011111
(End)

Michael S. Branicky, Table of n, a(n) for n = 1..81

A subset of A044813 (distinct run-lengths) and of A175413 (distinct runs).
These are the positions of first appearances in A165413.
The version for runs instead of run-lengths is A350952, firsts of A297770.
A000120 counts binary weight.
A005811 counts runs in binary expansion.
A242882 counts compositions with distinct multiplicities.
A318928 gives runs-resistance of binary expansion.
A351014 counts distinct runs in standard compositions.
Counting words with all distinct run-lengths:
- A032020 = binary expansions, for runs A351018.
- A329739 = compositions, for runs A351013.
- A351017 = binary words, for runs A351016.
- A351292 = patterns, for runs A351200.
Cf. A003242, A070939, A085207, A098859, A325545, A328592, A351015, A351202, A351203, A351291.

g[n_] := Table[ {Table[1, {i}], Table[0, {n - i + 1}]}, {i, Floor[(n + If[ OddQ@n, 1, 0])/2]}]; f[n_] := FromDigits[ If[ OddQ@n, Flatten@ Most@ Flatten[ g@n, 1], Flatten@ g@n], 2]; Array[f, 14]
s=Table[Length[Union[Length/@Split[IntegerDigits[n,2]]]],{n,0,1000}]; Table[Position[s,k][[1,1]]-1,{k,Union[s]}] (* Gus Wiseman, Feb 21 2022 *)

def a(n): # returns term by construction
    if n == 1: return 1
    q, r = divmod(n+1, 2)
    s = "".join("1"*i + "0"*(n+1-i) for i in range(1, q+1))
    if r == 0: s = s.rstrip("0")
    return int(s, 2)
print([a(n) for n in range(1, 16)]) # Michael S. Branicky, Feb 22 2022

a(15) and beyond from Michael S. Branicky, Feb 22 2022

A351205 Numbers whose binary expansion does not have all distinct runs.

Original entry on oeis.org

5, 9, 10, 17, 18, 20, 21, 22, 26, 27, 33, 34, 36, 37, 40, 41, 42, 43, 45, 46, 51, 53, 54, 58, 65, 66, 68, 69, 72, 73, 74, 75, 76, 77, 80, 81, 82, 83, 84, 85, 86, 87, 89, 90, 91, 93, 94, 99, 100, 101, 102, 105, 106, 107, 108, 109, 110, 117, 118, 119, 122, 129

Offset: 1

Gus Wiseman, Feb 07 2022

			The terms together with their binary expansions begin:
      5:     101     41:  101001     74: 1001010
      9:    1001     42:  101010     75: 1001011
     10:    1010     43:  101011     76: 1001100
     17:   10001     45:  101101     77: 1001101
     18:   10010     46:  101110     80: 1010000
     20:   10100     51:  110011     81: 1010001
     21:   10101     53:  110101     82: 1010010
     22:   10110     54:  110110     83: 1010011
     26:   11010     58:  111010     84: 1010100
     27:   11011     65: 1000001     85: 1010101
     33:  100001     66: 1000010     86: 1010110
     34:  100010     68: 1000100     87: 1010111
     36:  100100     69: 1000101     89: 1011001
     37:  100101     72: 1001000     90: 1011010
     40:  101000     73: 1001001     91: 1011011
For example, 77 has binary expansion 1001101, with runs 1, 00, 11, 0, 1, which are not all distinct, so 77 is in the sequence.

Mathematics Stack Exchange, What is a sequence run? (answered 2011-12-01)

Runs in binary expansion are counted by A005811, distinct A297770.
The complement is A175413, for run-lengths A044813.
The version for standard compositions is A351291, complement A351290.
A000120 counts binary weight.
A011782 counts integer compositions.
A242882 counts compositions with distinct multiplicities.
A318928 gives runs-resistance of binary expansion.
A325545 counts compositions with distinct differences.
A333489 ranks anti-runs, complement A348612, counted by A003242.
A334028 counts distinct parts in standard compositions.
A351014 counts distinct runs in standard compositions.
Counting words with all distinct runs:
- A351013 = compositions, for run-lengths A329739.
- A351016 = binary words, for run-lengths A351017.
- A351018 = binary expansions, for run-lengths A032020.
- A351200 = patterns, for run-lengths A351292.
- A351202 = permutations of prime factors.
Cf. A070939, A085207, A098859, A233564, A238130 or A238279, A283353, A328592, A350952, A351015, A351203.

q:= proc(n) uses ListTools; (l-> is(nops(l)<>add(
      nops(i), i={Split(`=`, l, 1)}) +add(
      nops(i), i={Split(`=`, l, 0)})))(Bits[Split](n))
    end:
select(q, [$1..200])[];  # Alois P. Heinz, Mar 14 2022

Select[Range[0,100],!UnsameQ@@Split[IntegerDigits[#,2]]&]

from itertools import groupby, product
def ok(n):
    runs = [(k, len(list(g))) for k, g in groupby(bin(n)[2:])]
    return len(runs) > len(set(runs))
print([k for k in range(130) if ok(k)]) # Michael S. Branicky, Feb 09 2022

cp's OEIS Frontend

A351592 Number of Look-and-Say partitions (A239455) of n without distinct multiplicities, i.e., those that are not Wilf partitions (A098859).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

Extensions

A165933 Least integer, k, whose value is n in A165413.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Python

Extensions

A351205 Numbers whose binary expansion does not have all distinct runs.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Python