OFFSET
1,6
COMMENTS
Every row in the table 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 n (first column in the triangle, see also A342594) whose symmetric representation of sigma(n) has width pattern w.
Not every possible Dyck path (see A001405) is realized as the width pattern in the symmetric representation of sigma. The (lexicographically) smallest such pattern w = ( 1 0 1 2 1 2 1 0 1 ) of length 9 would have to be realized by a number n = 2^s * p^4, s >= 0 and p an odd prime, that has 5 odd divisors; however pattern w requires 2^(s+1) < p and 2^(s+1) * p > p^2.
There are only k distinct width patterns of length 2*k - 1 when k is an odd prime. The values in the left parts of the k patterns are: w_i(j) = j, 1 <= j <= i <= k; w_i(j) = i-1, i+1 <= j <= k and j - i odd; w_i(j) = i, i+1 <= j <= k and j - i even. This follows from the fact that when for number n = 2^s * p^(k-1) with s >= 0 and k prime, and the leftmost maximum width in pattern w is m then p^m < 2^(s+1) < p^(m+1) so that the width pattern m m-1 m m-1 ... must continue to the center of w.
EXAMPLE
The first 7 rows in the irregular triangle below correspond to rows 1, 3, 6, 9, 15, 18 and 21 of the triangle in A249223 of widths. Collapsing each constant section of a row to a single value in A249223 produces the left half of the corresponding row in the triangle of this sequence.
Every power of 2 has the width pattern (1); every odd prime has the width pattern (1, 0, 1); the triangle in A249223 shows that 12 and 24 have the same width pattern ( 1 2 1 ) as number 6.
The first 21 distinct width patterns. The smallest number whose symmetric representation of sigma has the width pattern of that row is listed in the first column. All possible symmetric Dyck path patterns of lengths 1, 3, 5 and 7 are 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
15 1 0 1 2 1 0 1
18 1 2 1 2 1
21 1 0 1 0 1 0 1
30 1 2 1 2 1 2 1
45 1 0 1 2 1 2 1 2 1 0 1
60 1 2 3 2 3 2 1
63 1 0 1 0 1 2 1 0 1 0 1
72 1 2 3 2 1
75 1 0 1 2 1 0 1 2 1 0 1
78 1 2 1 0 1 2 1
81 1 0 1 0 1 0 1 0 1
90 1 2 1 2 3 2 3 2 1 2 1
105 1 0 1 2 1 2 1 0 1 2 1 2 1 0 1
120 1 2 3 4 3 2 1
135 1 0 1 2 1 2 1 2 1 2 1 2 1 0 1
147 1 0 1 0 1 0 1 0 1 0 1
150 1 2 1 2 1 2 1 2 1 2 1
MATHEMATICA
(* function a341969[ ] is defined in A341969*)
a342592[n_] := Module[{listW={}, k, w}, For[k=1, k<=n, k++, w=a341969[k]; If[!MemberQ[listW, w], AppendTo[listW, w]]]; Flatten[listW]]
a342592[81] (* 15 distinct patterns; the 15th is encountered at n=81 *)
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Hartmut F. W. Hoft, Mar 16 2021
STATUS
approved