Alternating permutation

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Not to be confused with alternating group.

In combinatorial mathematics, an alternating permutation of the set {1, 2, 3, ..., n} is an arrangement of those numbers into an order c1, ..., cn such that no element ci is between ci − 1 and ci + 1 for any value of i and c1< c2. In other words, ci < ci+ 1 if i is odd and ci > ci+ 1 if i is even. For example, the five alternating permutations of {1, 2, 3, 4} are:

  • 1, 3, 2, 4       because       1 < 3 > 2 < 4
  • 1, 4, 2, 3       because       1 < 4 > 2 < 3
  • 2, 3, 1, 4       because       2 < 3 > 1 < 4
  • 2, 4, 1, 3       because       2 < 4 > 1 < 3
  • 3, 4, 1, 2       because       3 < 4 > 1 < 2

This type of permutation was first studied by Désiré André in the 19th century.[1]

If the condition c1 < c2 is dropped, so we only require that no element ci is between ci − 1 and ci + 1, then the permutation is called a zigzag permutation. By exchanging 1 with n, 2 with n − 1, etc., each zigzag permutation with c1> c2 can be paired uniquely with an alternating permutation.

Related integer sequences[edit]

The determination of the number, An, of alternating permutations of the set {1, ..., n} is called André's problem. If Zn denotes the number of zigzag permutations of {1, ..., n} then it is clear from the pairing given above that Zn = 2An for n ≥ 2. The numbers An are known as Euler zigzag numbers or up/down numbers. The first few values of An are 1, 1, 1, 2, 5, 16, 61, 272, 1385, 7936, 50521, ... (sequence A000111 in OEIS). The first few values of Zn are 1, 1, 2, 4, 10, 32, 122, 544, 2770, 15872, 101042, ... (sequence A001250 in OEIS).

The numbers A2n with even indices are called secant numbers or zig numbers. The first few values are 1, 1, 5, 61, 1385, 50521, ... (sequence A000364 in OEIS). They appear as numerators in the Maclaurin series of sec x. Specifically,

\sec x = 1 + \frac{1}{2!}x^2 + \frac{5}{4!}x^4 + \cdots = \sum_{n=0}^\infty A_{2n} {x^{2n} \over ({2n})!}.

Secant numbers are related to Euler numbers by the formula E2n = (−1)nA2n. (En = 0 when n is odd.)

Correspondingly, the numbers A2n+1 with odd indices are called tangent numbers or zag numbers. The first few values are 1, 2, 16, 272, 7936, ... (sequence A000182 in OEIS). They appear as numerators in the Maclaurin series of tan x. Specifically,

\tan x = \frac{1}{1!}x + \frac{2}{3!}x^3 + \frac{16}{5!}x^5 + \cdots = \sum_{n=0}^\infty A_{2n+1} {x^{2n+1} \over ({2n+1})!}.

Tangent numbers are related to Bernoulli numbers by the formula

B_{2n} =(-1)^{n-1}\frac{2n}{4^{2n}-2^{2n}} A_{2n-1}

for n > 0.

Adding these series together gives the exponential generating function of the sequence An:

\sum_{n=0}^\infty A_n {x^n \over n!} = \sec x + \tan x = \tan\left({x \over 2} + {\pi \over 4}\right).
A_n = i^{n+1}\sum _{k=1}^{n+1} \sum _{j=0}^k {k\choose{j}} \frac{(-1)^j(k-2j)^{n+1}}{2^ki^kk}

André's theorem[edit]

In some contexts, one defines the terms alternating permutation and reverse-alternating permutation so that the former are arrangements of the set { 1, 2, 3, ..., n} into a sequence a1, ..., an satisfying

 a_1 > a_2 < a_3 > a_4 < \cdots \,

and the latter satisfy

 a_1 < a_2 > a_3 < a_4 > \cdots. \,

(A bijection between alternating and reverse-alternating permutations is given by

 a_i \mapsto n+1-a_i.) \,

Let En be the number of alternating permutations of the set { 1, 2, 3, ..., n}. The first several of these numbers are

 E_0 = 1,\quad E_1=1,\quad E_2=1,\quad E_3=2,\quad E_4=5,\quad E_5=16,\quad E_6=61,\quad E_7=272,\quad \ldots

André's theorem states:

The exponential generating function of the numbers En is
 f(x) = \sum_{n=0}^\infty E_n \frac{x^n}{n!} = \sec x + \tan x.

The radius of convergence of this series can be seen to be positive by noticing that En is bounded above by n!. In fact, the radius is π/2.

Proof[edit]

Here we prove André's theorem by means of a combinatorial argument showing that this generating function satisfies a certain differential equation, and we use the initial condition ƒ(0) = 1. This proof is due to André (1881) and also appears in Stanley (1950, pages 46–7). See also this preprint by Stanley.

The principal combinatorial result that we need is this identity:

 2E_{n+1} = \sum_{k=0}^n \binom n k E_k E_{n-k}\text{ if }n\ge 1.\qquad\qquad(1)

The proviso that n ≥ 1 is crucial.

A proof of this combinatorial identity runs as follows. To choose an alternating or reverse-alternating permutation of the set { 1, 2, 3, ..., nn + 1 }, we

  • choose a subset of size k of the set { 1, ..., n }, then
  • choose a reverse-alternating permutation a1, ..., ak of the set { 1, ..., k }, then
  • choose a reverse-alternating permutation b1, ..., bnk of the set { k + 1, ..., n }.

Then the permutation

 a_k, a_{k-1}, a_{k-2},\ldots, a_1, n+1, b_1, b_2, b_3,\ldots b_{n-k} \,

is either alternating or reverse-alternating. The number of ways to choose a permutation of { 1, 2, 3, ..., nn + 1 } that is either alternating or reverse-alternating is En+1, and the number of ways to complete the sequence of steps above is

 \binom n k E_k E_{n-k}. \,

This needs to be done for each possible value of k to get a complete list, hence we sum from k = 0 to k = n. This completes the proof of the identity (1).

Multiplication of both sides of (1) by xn+1/(n+1)! and summing over n ≥ 1, and then prepending the constant and first-degree terms, yields

 2f(x) = 2E_0 + 2E_1 x + 2\sum_{n=1}^\infty E_{n+1}\frac{x^{n+1}}{(n+1)!} = 2E_0 + 2 E_1 x + \sum_{n=1}^\infty \sum_{k=0}^n \binom n k E_k E_{n-k} \frac{x^{n+1}}{(n+1)!}.

Differentiating both sides, we get

 2f'(x) = 2E_1 + 2\sum_{n=1}^\infty E_{n+1}\frac{x^n}{n!} = 2E_1 + \sum_{n=1}^\infty \sum_{k=0}^n \binom n k E_k E_{n-k} \frac{x^n}{n!}.

In the last sum, the index n goes from 1 to ∞, not from 0 to ∞. If we change the lower bound of summation from 1 to 0, we simply add 1 to the sum, and compensate by changing the initial term, 2E1 = 2, to E1 = 1, getting

 E_1 + \sum_{n=0}^\infty \sum_{k=0}^n \binom n k E_k E_{n-k} \frac{x^n}{n!} = 1 + \sum_{n=0}^\infty \sum_{k=0}^n \left(E_k \frac{x^k}{k!}\right) \left( E_{n-k} \frac{x^{n-k}}{(n-k)!}\right).

The last sum is over all pairs of positive integers, so the expression above can be rearranged as

 1 + \sum_{i=0}^\infty \sum_{j=0}^\infty E_i\frac{x^i}{i!}\cdot E_j \frac{x^j}{j!}

(see Cauchy product).

The expression  E_i\frac{x^i}{i!} does not change as j goes from 0 to ∞; hence it can be pulled out of the inside sum, getting

 1 + \sum_{i=0}^\infty\left( E_i\frac{x^i}{i!} \sum_{j=0}^\infty E_j\frac{x^j}{j!} \right).

Now the second sum does not change as i goes from 0 to ∞; hence it can be pulled out of the outer sum, getting

 1 + \left(\sum_{j=0}^\infty E_j\frac{x^j}{j!} \right)\left(\sum_{i=0}^\infty E_i\frac{x^i}{i!} \right).

This is

 1 + (f(x))^2. \,

Consequently we have a differential equation

 2f'(x) = 1 + (f(x))^2. \,

This can be solved by separation of variables, getting

 f(x) = \tan\left( \frac x 2 + \text{constant} \right).

We have an initial condition ƒ(0) = 1, so we have

 f(x) = \tan\left( \frac x 2 + \frac\pi4 \right).

Finally, a standard tangent half-angle trigonometric identity gives us

 f(x) = \sec x + \tan x. \,

This completes the proof of André's theorem.

See also[edit]

Citations[edit]

  1. ^ Jessica Millar, N. J. A. Sloane, Neal E. Young, "A New Operation on Sequences: the Boustrouphedon Transform" J. Combinatorial Theory, Series A 76(1):44–54 (1996)

References[edit]

  • André, D. "Développements de sec x et tan x." Comptes Rendus Acad. Sci., Paris 88, 965–967, 1879.
  • André, Désiré (1881), Sur les permutations alternées, Journal de mathématiques pures et appliquées, 3e série 7: 167–84 
  • Stanley, Richard P. (2009). "A Survey of Alternating Permutations". arXiv:0912.4240 [math.CO].
  • Stanley, Richard P. (2011). Enumerative Combinatorics, Volume I, second edition. Cambridge University Press. .

Further reading[edit]

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