OFFSET
1
COMMENTS
Let s = (s(n)) be a strictly increasing sequence of positive integers with infinite complement, t = (t(n)).
For n >=1, let s'(n) be the number of s(i) that are <= n-1 and let t'(n) be the number of t(i) that are <= n-1.
Define w(1) = 0, w(t(1)) = 1, and w(n) = 0w(s'(n)) if n is in s, and w(n) = 1w(t'(n)) if n is in t. Then (w(n)) is the "s-induced ordering" of all 01-words.
In the following list, W represents the sequence of words w(n) induced by A001651. The list includes five partitions and two permutations of the positive integers.
positions of 1-free words in W: A006999;
positions of 0-free words in W: A029858;
length of w(n): A342774;
positions in W of words w(n) such that # 0's = # 1's: A342775;
positions in W of words w(n) such that # 0's < # 1's: A342776;
positions in W of words w(n) such that # 0's > # 1's: A342777;
positions in W of words having last digit 0: A342778;
positions in W of words having last digit 1: A342779;
positions in W of words w(n) such that first digit = last digit: A342780;
positions in W of words w(n) such that first digit != last digit: A342781;
positions in W of words w(n) such that 1st digit = 0 and last digit 0: A342748;
positions in W of words w(n) such that 1st digit = 0 and last digit 1: A342783;
positions in W of words w(n) such that 1st digit = 1 and last digit 0: A342784;
positions in W of words w(n) such that 1st digit = 1 and last digit 1: A342785;
position in W of n-th positive integer (base 2): A342786;
positions in W of binary complement of w(n): A342787;
sum of digits in w(n): A342788;
number of runs in w(n): A342789;
positions in W of palindromes: A342790;
positions in W of words such that #0's - #1's is odd: A342791;
positions in W of words such that #0's - #1's is even: A342792.
position in W of the reversal of the n-th word in A342798.
For a guide to related sequences, see A341256.
EXAMPLE
The first 16 words w(n): 0, 00, 1, 000, 01, 10, 0000, 001, 100, 010, 00000, 11, 0001, 0100, 1000, 0010.
MATHEMATICA
z = 100; s = Table[Floor[(3 n - 1)/2], {n, 1, z}]; (* A001651 *)
t = Complement[Range[Max[s]], s]; (* A008585 *)
s1[n_] := Length[Intersection[Range[n - 1], s]];
t1[n_] := n - 1 - s1[n];
Table[s1[n], {n, 1, z}]; (* A004523 *)
Table[t1[n], {n, 1, z}]; (* A002264 *)
w[1] = {0}; w[t[[1]]] = {1};
w[n_] := If[MemberQ[s, n], Join[{0}, w[s1[n]]], Join[{1}, w[t1[n]]]];
tt = Table[w[n], {n, 1, z}] (* A342753, words *)
Flatten[tt] (* A342753, concatenated *)
Map[Length, tt] (* A342774 *)
Flatten[Position[Map[Union, tt], {0}]]; (* A006999 *)
Flatten[Position[Map[Union, tt], {1}]]; (* A029858 *)
zz = Range[Length[tt]];
Select[zz, Count[tt[[#]], 0] == Count[tt[[#]], 1] &] (* A342775 *)
Select[zz, Count[tt[[#]], 0] < Count[tt[[#]], 1] &] (* A342776 *)
Select[zz, Count[tt[[#]], 0] > Count[tt[[#]], 1] &] (* A342777 *)
Select[zz, Last[tt[[#]]] == 0 &] (* A342778 *)
Select[zz, Last[tt[[#]]] == 1 &] (* A342779 *)
Select[zz, First[tt[[#]]] == Last[tt[[#]]] &] (* A342780 *)
Select[zz, First[tt[[#]]] != Last[tt[[#]]] &] (* A342781 *)
Select[zz, First[tt[[#]]] == 0 && Last[tt[[#]]] == 0 &] (* A342782 *)
Select[zz, First[tt[[#]]] == 0 && Last[tt[[#]]] == 1 &] (* A342783 *)
Select[zz, First[tt[[#]]] == 1 && Last[tt[[#]]] == 0 &] (* A342784 *)
Select[zz, First[tt[[#]]] == 1 && Last[tt[[#]]] == 1 &] (* A342785 *)
d[n_] := If[First[w[n]] == 1, FromDigits[w[n], 2]];
Flatten[Table[Position[Table[d[n], {n, 1, 200}], n], {n, 1, 200}]] (* A342786 *)
comp = Flatten[Table[Position[tt, 1 - w[n]], {n, 1, 50}]] (* A342787 *)
Table[Total[w[n]], {n, 1, 100}] (* A342788 *)
Map[Length, Table[Map[Length, Split[w[n]]], {n, 1, 100}]] (* A342789 *)
Select[zz, tt[[#]] == Reverse[tt[[#]]] &] (* A342790 *)
Select[zz, OddQ[Count[w[#], 0] - Count[w[#], 1]] &] (* A342791 *)
Select[zz, EvenQ[Count[w[#], 0] - Count[w[#], 1]] &] (* A342792 *)
Flatten[Table[Position[tt, Reverse[w[n]]], {n, 1, 150}]]; (* A342798 *)
CROSSREFS
KEYWORD
nonn,base
AUTHOR
Clark Kimberling, Apr 10 2021
STATUS
approved