cp's OEIS Frontend

A342592 Irregular triangle read by rows in which row n lists the next width pattern not yet encountered as n increases in the symmetric representation of sigma(n), where each contiguous sequence of identical widths w in A249223 is replaced by a single entry of w (see A341969).

%I A342592 #12 Mar 18 2021 23:51:36
%S A342592 1,1,0,1,1,2,1,1,0,1,0,1,1,0,1,2,1,0,1,1,2,1,2,1,1,0,1,0,1,0,1,1,2,1,
%T A342592 2,1,2,1,1,0,1,2,1,2,1,2,1,0,1,1,2,3,2,3,2,1,1,0,1,0,1,2,1,0,1,0,1,1,
%U A342592 2,3,2,1,1,0,1,2,1,0,1,2,1,0,1,1,2,1,0,1,2,1,1,0,1,0,1,0,1,0,1
%N A342592 Irregular triangle read by rows in which row n lists the next width pattern not yet encountered as n increases in the symmetric representation of sigma(n), where each contiguous sequence of identical widths w in A249223 is replaced by a single entry of w (see A341969).
%C A342592 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.
%C A342592 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.
%C A342592 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.
%e A342592 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.
%e A342592 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.
%e A342592 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).
%e A342592     1  1
%e A342592     3  1 0 1
%e A342592     6  1 2 1
%e A342592     9  1 0 1 0 1
%e A342592    15  1 0 1 2 1 0 1
%e A342592    18  1 2 1 2 1
%e A342592    21  1 0 1 0 1 0 1
%e A342592    30  1 2 1 2 1 2 1
%e A342592    45  1 0 1 2 1 2 1 2 1 0 1
%e A342592    60  1 2 3 2 3 2 1
%e A342592    63  1 0 1 0 1 2 1 0 1 0 1
%e A342592    72  1 2 3 2 1
%e A342592    75  1 0 1 2 1 0 1 2 1 0 1
%e A342592    78  1 2 1 0 1 2 1
%e A342592    81  1 0 1 0 1 0 1 0 1
%e A342592    90  1 2 1 2 3 2 3 2 1 2 1
%e A342592   105  1 0 1 2 1 2 1 0 1 2 1 2 1 0 1
%e A342592   120  1 2 3 4 3 2 1
%e A342592   135  1 0 1 2 1 2 1 2 1 2 1 2 1 0 1
%e A342592   147  1 0 1 0 1 0 1 0 1 0 1
%e A342592   150  1 2 1 2 1 2 1 2 1 2 1
%t A342592 (* function a341969[ ] is defined in A341969*)
%t A342592 a342592[n_] := Module[{listW={}, k, w}, For[k=1, k<=n, k++, w=a341969[k]; If[!MemberQ[listW, w], AppendTo[listW, w]]]; Flatten[listW]]
%t A342592 a342592[81] (* 15 distinct patterns; the 15th is encountered at n=81 *)
%Y A342592 Subsequence of A341969.
%Y A342592 Cf. A001405, A235791, A237048, A237270, A237591, A237593, A249223, A342594.
%K A342592 nonn,tabf
%O A342592 1,6
%A A342592 _Hartmut F. W. Hoft_, Mar 16 2021