Boustrophedon transform: Encyclopedia - Boustrophedon transform

Boustrophedon transform

In mathematics, the boustrophedon transform is a procedure which maps one sequence to another. The transformed sequence is computed by filling a triangle in boustrophedon (zig-zag) manner.

Boustrophedon transform - Definition

Given a sequence , the boustrophedon transform yields another sequence, say , which is constructed by filling up a triangle as pictured on the right. Number the rows in the triangle starting from 0, and fill the rows consecutively. Let k denote the number of the row currently being filled.

If k is odd, then put the number a_k on the right end of the row and fill the row from the right to the left, with every entry being the sum of the number to the right and the number to the upper right. If k is even, then put the number a_k on the left end and fill the row from the left to the right, with every entry being the sum of the number to the left and the number to the upper left.

The numbers b_k forming the transformed sequence can then be found on the left end of odd-numbered rows and on the right end of even-numbered rows, that is, opposite to the numbers a_k.

Boustrophedon transform - Recurrence relation

A more formal definition uses a recurrence relation. Define the numbers T_k,n (with k ≥ n ≥ 0) by

Then the transformed sequence is defined by b_n = T_n,n.

Boustrophedon transform - The up/down numbers

The up/down numbers u_n count the number of permutations of the set {1, 2, 3, …, n} which alternately rise and fall, starting with a rise. For example, u₄ = 5, because there are five permutations of {1, 2, 3, 4} satisfying these conditions, namely:

• 1, 3, 2, 4
• 1, 4, 2, 3
• 2, 3, 1, 4
• 2, 4, 1, 3
• 3, 4, 1, 2

The sequence of up/down numbers is the boustrophedon transform of the unit sequence

For this reason, the up/down numbers are also called the boustrophedon transform numbers. Yet another name is Euler numbers, though this name is usually reserved for a slightly different sequence, as explained in Euler numbers.

Boustrophedon transform - The exponential generating function

The exponential generating function of a sequence (a_n) is defined by

The exponential generating function of the boustrophedon transform (b_n) is related to that of the original sequence (a_n) by

The exponential generating function of the unit sequence is 1, so that of the up/down numbers is

This explains why the up/down numbers appear as the Taylor coefficients of the tangent and secant functions.

Adapted from the Wikipedia article "Boustrophedon transform", under the G.N U Free Docmentation License. Please also see http://en.wikipedia.org/wiki