cp's OEIS Frontend

A374411 Triangle T(n, k) read by rows: Maximum number of linear patterns of length k in a circular permutation of length n taken from row n in A194832.

%I A374411 #19 Jul 19 2024 22:34:41
%S A374411 1,1,2,1,2,3,1,2,6,4,1,2,6,16,5,1,2,6,20,25,6,1,2,6,24,60,36,7,1,2,6,
%T A374411 24,85,126,49,8,1,2,6,24,100,222,196,64,9,1,2,6,24,115,390,511,288,81,
%U A374411 10,1,2,6,24,120,558,1085,912,405,100,11,1,2,6,24,120,654,1911,2328,1458,550,121,12
%N A374411 Triangle T(n, k) read by rows: Maximum number of linear patterns of length k in a circular permutation of length n taken from row n in A194832.
%C A374411 Pattern counting considers only one revolution otherwise every sufficiently long circular permutation, with enough revolutions allowed, contains every pattern.
%C A374411 Each column k is divisible by k, because as we count linear patterns inside a circular permutation, we may obtain all circular shifts of the subset which represents a particular pattern.
%F A374411 T(n, k+1)/(k+1) <= A371823(n-1, k) <= A373778(n-1, k).
%e A374411 The triangle begins:
%e A374411    n| k: 1| 2| 3|  4|   5|   6|   7|  8|  9
%e A374411   =========================================
%e A374411   [1]    1
%e A374411   [2]    1, 2
%e A374411   [3]    1, 2, 3
%e A374411   [4]    1, 2, 6,  4
%e A374411   [5]    1, 2, 6, 16,   5
%e A374411   [6]    1, 2, 6, 20,  25,   6
%e A374411   [7]    1, 2, 6, 24,  60,  36,   7
%e A374411   [8]    1, 2, 6, 24,  85, 126,  49,  8
%e A374411   [9]    1, 2, 6, 24, 100, 222, 196, 64, 9
%e A374411 .
%e A374411 Row 5 of A194832 is [3, 1, 4, 2, 5].
%e A374411 T(5, 4) = 16 because we will find these 16 distinct patterns of length 4:
%e A374411    [3, 1, 4, 2] [1, 4, 2, 3] [4, 2, 3, 1] [2, 3, 1, 4]
%e A374411  These are rotations of the ordering [1, 4, 2, 3].
%e A374411    [1, 4, 2, 5] [4, 2, 5, 1] [2, 5, 1, 4] [5, 1, 4, 2]
%e A374411  These are rotations of the ordering [1, 3, 2, 4].
%e A374411    [2, 5, 3, 1] [5, 3, 1, 2] [3, 1, 2, 5] [1, 2, 5, 3]
%e A374411  These are rotations of the ordering [1, 2, 4, 3].
%e A374411    [5, 3, 1, 4] [3, 1, 4, 5] [1, 4, 5, 3] [4, 5, 3, 1]
%e A374411  These are rotations of the ordering [1, 3, 4, 2].
%Y A374411 Cf. A194832, A371823, A373778.
%K A374411 nonn,tabl
%O A374411 1,3
%A A374411 _Thomas Scheuerle_, Jul 08 2024