Entringer numbers E(n,k): count alternating (zigzag) permutations of {1,...,n} beginning with k+1. Recurrence: E(n,k)=E(n,k-1)+E(n-1,n-k), E(0,0)=1. The diagonal E(n,n) gives the sequence 1,1,1,2,5,16,61,272... (Euler zigzag numbers).
How to Use
Enter n:
- Triangle: Full E(n,k)
- Diagonal: Zigzag numbers
- Sums: Euler numbers
Boustrophedon
The triangle is built 'boustrophedon' (ox-turning): odd rows left-to-right, even rows right-to-left. Each entry = previous + entry above from opposite end. This zigzag construction mirrors the alternating permutation definition.
Connections
E(2n,2n) = E_{2n} = secant numbers (Euler numbers). E(2n+1,2n+1) = T_n = tangent numbers. The Entringer triangle unifies both into a single combinatorial object.
Step-by-Step Instructions
- 1Enter n.
- 2View triangle.
- 3See diagonal.
- 4Connect to Euler.
- 5Explore recurrence.