OEIS/Triangles

• A282698 contains A003121 (for triangles with strict "less than" interlacing), and 20, 1744
• A003121

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