A342595 Irregular triangle of A342592 read by rows arranged first by length of the width pattern and then lexicographically within blocks of patterns of equal length.

1, 1, 0, 1, 1, 2, 1, 1, 0, 1, 0, 1, 1, 2, 1, 2, 1, 1, 2, 3, 2, 1, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 1, 2, 1, 0, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 3, 2, 3, 2, 1, 1, 2, 3, 4, 3, 2, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 2, 3, 2, 3, 2, 3, 2, 1

Offset: 1

Author

Hartmut F. W. Hoft, Mar 16 2021

Keywords

• nonn
• tabf

Comments

Let u and v be two symmetric width patterns; then u < v if u is shorter than v, and if they have the same length then they are ordered lexicographically, i.e., if i is the first index where u and v differ and u(i) < v(i) then u < v.

This sequence is a permutation of the rows of the irregular triangle in A342592. Every row in the triangle representing a width pattern w contains an odd number 2*k - 1, k >= 1, of entries where k is the number of odd divisors of the smallest number whose symmetric representation of sigma realizes that pattern.

The number of distinct width patterns w of the same length 2*k-1 created by numbers with k odd divisors is computationally challenging since the numbers of their first occurrence can be very large (A342592, A342596). The counts in the table below are already established for n <= 5*10^5 and have not changed through 10^7; the counts are not stable at that larger level for width patterns of numbers with more than 8 odd divisors:

# odd divisors 1 2 3 4 5 6 7 8

pattern count 1 2 3 6 5 16 7 40

A001405 1 2 3 6 10 20 35 70

For any odd number q with k divisors and 2^s < q < 2^(s+1), s >= 0, any number 2^t * q with t > s has the lexicographically largest symmetric width pattern 1 2 3 ... k-2 k-1 k k-1 k-2 ... 3 2 1 of length 2*k - 1. Therefore, the sequence q, 2 * q, 2^2 * q, ... , 2^s * q instantiates at most s+1 different symmetric width patterns; these range from 2 for prime numbers q, patterns (1 0 1) and (1 2 1), to the maximum of s+1 different patterns such as for q = 105 = 3*5*7.

Examples

			The number of entries through the center in a row of the triangle below equals the number of odd divisors of any number that has that pattern of widths.
The pattern in row 10 of the triangle below, realized first by n = 30 which labels the row is the smallest number with width pattern (1 2 1 2 1 2 1); 42, 54 and 66 are the other numbers less than 100 realizing that pattern.
The triangle below lists the first 21 distinct symmetric width patterns in the order described above. The smallest number whose symmetric representation of sigma has the width pattern of that row is listed as first column (see A342596). All possible symmetric width patterns of lengths 1, 3, 5 and 7 are realized in the triangle below; their respective counts are A001405(1,2,3,4) = (1,2,3,6).
     1   1
     3   1 0 1
     6   1 2 1
     9   1 0 1 0 1
    18   1 2 1 2 1
    72   1 2 3 2 1
    21   1 0 1 0 1 0 1
    15   1 0 1 2 1 0 1
    78   1 2 1 0 1 2 1
    30   1 2 1 2 1 2 1
    60   1 2 3 2 3 2 1
   120   1 2 3 4 3 2 1
    81   1 0 1 0 1 0 1 0 1
   162   1 2 1 2 1 2 1 2 1
   648   1 2 3 2 3 2 3 2 1
  1296   1 2 3 4 3 4 3 2 1
  5184   1 2 3 4 5 4 3 2 1
   147   1 0 1 0 1 0 1 0 1 0 1
    63   1 0 1 0 1 2 1 0 1 0 1
    75   1 0 1 2 1 0 1 2 1 0 1
    45   1 0 1 2 1 2 1 2 1 0 1
		

Crossrefs

Cf. A001405, A237593, A249223, A341969, A342592, A342594, A342596.

Programs

Mathematica

(* function a341969[ ] is defined in A341969 *)
lexicographic[s1_, s2_] := Module[{k=1}, While[s1[[2, k]]==s2[[2, k]], k++]; s1[[2,k]]