# OEIS/Triangles

From teherba.org

< OEIS

Number of ways to arrange the numbers 1,2,...,n(n+1)/2 in a triangle so that the rows interlace; e.g. one of the 12 triangles counted by a(4) is 6 4 8 2 5 9 1 3 7 10 - Clark Kimberling, Mar 25 2012 The a(4) = 12 ways to fill a triangle with the numbers 0 through 9: 5 6 6 5 3 7 3 7 2 7 2 7 1 4 8 1 4 8 1 4 8 1 4 8 0 2 6 9 0 2 5 9 0 3 5 9 0 3 6 9 5 3 3 4 3 6 2 6 2 7 3 7 1 4 8 1 5 8 1 5 8 1 5 8 0 2 7 9 0 4 7 9 0 4 6 9 0 2 6 9 4 4 5 4 2 6 2 7 2 6 3 6 1 5 8 1 5 8 1 4 8 1 5 8 0 3 7 9 0 3 6 9 0 3 7 9 0 2 7 9 - R. H. Hardin, Jul 06 2012

- A231074 The number of possible ways to arrange the sums x_i + x_j (1 <= i < j <= n) of the items x_1 < x_2 <...< x_n in nondecreasing order: 1, 1, 1, 1, 2, 12, 244. nonn,more. Vladimir Letsko, Mathematical Marathon, Problem 183 (in Russian)
- A231085 The number of possible ways to arrange the sums x_i + x_j (1 <= i < j <= n) of the items x_1 < x_2 <...< x_n in increasing order provided that all sums are different: 1, 1, 1, 1, 2, 12, 168, 4676. nonn,more. Vladimir Letsko, Mathematical Marathon, Problem 183 (in Russian). For n<=5, a(n) = A231074(n), but for n>5, a(n) < A231074(n). For instance, let n = 6 and a < b < c < d < e < f. Then the arrangement a+b <= a+c <= a+d <= a+e <= b+c <= b+d <= a+f <= b+e <= b+f <= c+d <= c+e <= c+f <= d+e <= d+f <= e+f is possible (e.g., for a = 1, b = 5, c = 9, d = 12, e=13, f = 16), while the same arrangement with "<" instead of "<=" is not possible.
- A131811 Number of symbolic sequences on n symbols that can be realized by the arrangement of the real roots of some polynomial of degree n and its derivatives: 1, 1, 2, 10, 116. nonn,hard,more
- A213457 Intertwining numbers. (Formerly M1988): 1, 1, 2, 10, 148, 7384, 1380960, more. a(4)=10 for example is the number of ways of arranging 1 a, 2 b's, 3 c's and 4 d's so that if we look at any two letters, i and j say, with i<j, then any pair of i's are separated by at least one j.
- A083568 Duplicate of A003121. dead