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. Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations related to: Mathematics Look up Mathematics on Wiktionary, the free dictionary Wikimedia Commons has media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ... This is a page about mathematics. ... Boustrophedon is an ancient way of writing manuscripts and other inscriptions in which, rather than going from left to right as in modern English, or right to left as in Arabic, alternate lines must be read in opposite directions. ...

Definition

The boustrophedon transform: Start with the original sequence (in blue), then add numbers as indicated by the arrows, and finally read of the transformed sequence on the other side (in red).

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.

Recurrence relation

A more formal definition uses a recurrence relation. Define the numbers T_k,n (with k ≥ n ≥ 0) by Recurrent redirects here; for the meaning of recurrent in contemporary hit radio, see Recurrent rotation. ...

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

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: In mathematics, especially in abstract algebra and related areas, a permutation is a bijection from a finite set X onto itself. ...

• 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. The Euler numbers are a sequence En of integers defined by the following Taylor series expansion: (Note that e, the base of the natural logarithm, is also occasionally called Eulers number, as is the Euler characteristic. ...

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.

References

• Jessica Millar, N.J.A. Sloane, Neal E. Young, "A New Operation on Sequences: the Boustrouphedon Transform," Journal of Combinatorial Theory, Series A, volume 76, number 1, pages 44–54, 1996. Also available in a slightly different version as e-print math.CO/0205218 on the arXiv.

The title given to this article is incorrect due to technical limitations. ...

External links

• Sequence A000111 on the On-Line Encyclopedia of Integer Sequences contains the up/down numbers.