A New Universal Cycle for Permutations

Abstract

We introduce a novel notation, the relaxed shorthand notation, to encode permutations. We then present a simple shift rule that exhaustively lists out each of the permutations exactly once. The shift rule induces a cyclic Gray code for permutations where successive strings differ by a rotation or a shift. By concatenating the first symbol of each string in the listing, we produce a universal cycle for permutations in relaxed shorthand notation. We also prove that the universal cycle can be constructed in O(1)-amortized time per symbol using O(n) space.

1 Universal Cycles for Permutations

A universal cycle for a set $\mathbf {S}$ is a cyclic sequence of length $|\mathbf {S}|$ whose substrings of length n encode $|\mathbf {S}|$ distinct objects in $\mathbf {S}$. As an example, the cyclic sequence 112132233 is a universal cycle for the set of 3-ary strings of length 2; the 9 unique substrings of length 2 when considered cyclicly are:

$$\begin{aligned} 11, 12, 21, 13, 32, 22, 23, 33, 31. \end{aligned}$$

A permutation of a character set $\langle n\rangle = \{1, 2, \ldots , n\}$ is an ordered arrangement of n distinct symbols in $\langle n\rangle$. A universal cycle for permutations of order n is a cyclic sequence of length n! whose substrings encode each of the permutations of $\langle n\rangle$ exactly once. Universal cycles for permutations do not exist under standard one-line notation when $n > 2$ [1]. To demonstrate the non-existence result, suppose there exists a universal cycle for permutations in one-line notation, then the universal cycle must contain the substring $n (n-1) \cdots 1$. The next length n substring of the universal cycle starts with $(n-1) (n-2) \cdots 1$, and must end with the symbol n. By similar reasoning, the next n symbols after $n (n-1) \cdots 1$ are exactly $n (n-1) \cdots 1$, a repetition and thus contradicts with the assumption that it is a universal cycle.

Several other notations were introduced to construct universal cycles for permutations. Jackson proved that universal cycles for k-permutations of $\langle n\rangle$ exist when $k < n$, where a k-permutation is an ordered arrangement of k distinct symbols in $\langle n\rangle$ [4]. Knuth extended Jackson’s result by introducing shorthand notation to encode permutations [6]. The shorthand notation of a permutation $a_1 a_2 \cdots a_n$ is $a_1 a_2 \cdots a_{n-1}$. A shorthand universal cycle for permutations of order n is a cyclic sequence of length n! that contains each of the unique permutations of $\langle n\rangle$ in shorthand notation as a substring exactly once. For example, the cyclic sequence

$$\begin{aligned} 123413243214213423143124 \end{aligned}$$

is a shorthand universal cycle for permutations of order 4; the 24 unique permutations when considered cyclicly are:

$$\begin{aligned}&123\underline{4}, 234\underline{1}, 341\underline{2}, 413\underline{2}, 132\underline{4}, 324\underline{1}, 243\underline{1}, 432\underline{1}, \\&321\underline{4}, 214\underline{3}, 142\underline{3}, 421\underline{3}, 213\underline{4}, 134\underline{2}, 342\underline{1}, 423\underline{1}, \\&231\underline{4}, 314\underline{2}, 143\underline{2}, 431\underline{2}, 312\underline{4}, 124\underline{3}, 241\underline{3}, 412\underline{3}. \end{aligned}$$

The last symbol of each permutation in the listing is determined by a length 3 substring of the shorthand universal cycle. Holroyd, Ruskey and Williams provided efficient constructions to generate shorthand universal cycles for permutations in O(1)-amortized time per symbol and O(1)-amortized time per n symbols respectively using O(n) space [2, 3, 7]. Permutations have also been encoded using relative order [1]. For example, 321341 is an order-isomorphic universal cycle for permutations of order 3 since its substrings are order-isomorphic to 321, 213, 123, 231, 312, 132. Johnson verified a conjecture in [1] to show that order-isomorphic universal cycles for permutations exist using only $n+1$ symbols [5]. However, there is currently no known efficient construction to generate order-isomorphic universal cycles for permutations for all order n.

In this paper, a novel notation, the relaxed shorthand notation, is introduced to represent permutations. We then present a shift-based construction for producing a universal cycle for permutations in relaxed shorthand notation. The construction is based on the following function over permutations, where k is the largest possible position such that $a_k > a_{k+1}$:

$$\begin{aligned}&{f}(a_1a_2\cdots a_n) = \left\{ \begin{array}{ll} a_2 a_1 a_3 a_4 \cdots a_n &{} \quad \text{ if } \; a_2 = 1 \quad \text{ and } \quad a_3 a_4 \cdots a_n \\ &{} \quad \text{ is } \text{ strictly } \text{ increasing; } \\ a_2 a_3 \cdots a_k a_1 a_{k+1} a_{k+2}\cdots a_{n} &{} \quad \text{ if } \; a_2 = 1 \quad \text{ and } \quad a_1 > a_{k+1}; \\ a_2 a_3 \cdots a_{k-1} a_1 a_k a_{k+1} \cdots a_{n} &{} \quad \text{ if } \; a_2 = 1 \quad \text{ and } \quad a_1 < a_{k+1}; \\ a_2 a_3 \cdots a_{n} {a_1} &{} \quad \text{ otherwise, } \end{array} \right. \end{aligned}$$

As an illustration, successive applications of this rule for $n = 4$ starting with the permutation 1234 produce the following listing:

$$\begin{aligned} \begin{aligned} 1234, 2341, 3412, 41{2}3, 1423, 4231, 2314, 3142, \\ 1432, 4321, 3214, 21{4}3, 1243, 2431, 4312, 31{2}4, \\ 1324, 3241, 2413, 4132, 1342, 3421, 4213, 21{3}4. \end{aligned} \end{aligned}$$

(1)

Observe that each permutation of length 4 is visited exactly once and that by applying one more application of the rule, we return to the first string 1234. This property holds in general for all $n \ge 1$. This leads to the following theorem, where ${\varPi }({n})$ denotes the set of permutations of $\langle n\rangle$.

Theorem 1

The shift rule f induces a cyclic ordering on ${\varPi }({n})$.

The rest of the paper is outlined as follows. In Sect. 2, we introduce the relaxed shorthand notation. In Sect. 3, we prove Theorem 1, which leads to a universal cycle for permutations in relaxed shorthand notation. Then in Sect. 4, we present an algorithm that generates a universal cycle for permutations in this new notation in O(1)-amortized time per symbol using O(n) space.

2 A Novel Notation to Represent Permutations

This section introduces the relaxed shorthand notation to represent permutations. The relaxed shorthand notation uses a length n string $\alpha = a_1 a_2 \cdots a_n$ with $n-1$ or n distinct symbols to represent a permutation of $\langle n\rangle$. If $\alpha$ contains n distinct symbols, then it simply represents the permutation $a_1 a_2 \cdots a_n$. Otherwise if $\alpha$ contains $n-1$ distinct symbols, then by pigeonhole principle there is a symbol which appears twice within $\alpha$. Let $a_i$ and $a_j$ be the same symbol within $\alpha$ such that $i < j$. We can then obtain a length $n-1$ string with $n-1$ distinct symbols $\beta$ by simply ignoring $a_j$, and shifting all symbols after $a_j$ to the left by one position, that is $\beta = a_1 a_2 \cdots a_{j-1} a_{j+1} \cdots a_n$ when $j < n$, and $\beta = a_1 a_2 \cdots a_{n-1}$ if $j = n$. We then treat $\beta$ as the shorthand notation of a permutation. Thus, the corresponding permutation can be obtained by appending the missing symbol in $\langle n\rangle$ to $\beta$. As an example, the permutation 1234 can be represented by 1123, 1213, 1223, 1231, 1232, 1233 and 1234 in relaxed shorthand notation.

A relaxed shorthand universal cycle for permutations of order n is a cyclic sequence of length n! that contains each of the unique permutations of $\langle n\rangle$ in relaxed shorthand notation as a substring exactly once. As an example, the cyclic sequence

$$\begin{aligned} 123414231432124313241342 \end{aligned}$$

is a relaxed shorthand universal cycle for permutations of order 4; the 24 unique permutations when considered cyclicly are listed out in (1).

Lemma 1

A shorthand universal cycle for $\mathbf {S} \in {\varPi }(n)$ is a relaxed shorthand universal cycle for $\mathbf {S}$.

Proof

A shorthand universal cycle for $\mathbf {S}$ has its length equal to $|\mathbf {S}|$ and contains each of the length $n-1$ prefixes of permutations in $\mathbf {S}$ as a substring exactly once. Let $\alpha = a_1 a_2 \cdots a_n$ be a permutation in $\mathbf {S}$. Observe that appending any symbol in $\langle n\rangle$ at the end of the length $n-1$ prefix of $\alpha$, that is $a_1 a_2 \cdots a_{n-1}$, produces a string that corresponds to $\alpha$ in relaxed shorthand notation. Thus, a shorthand universal cycle for $\mathbf {S}$ also contains each of the permutations in relaxed shorthand notation as a substring exactly once, and thus is a relaxed shorthand universal cycle for $\mathbf {S}$. $\square$

3 Proof of Theorem 1

In [8], Williams introduced the cool-lex ordering that exhaustively lists out multiset permutations. A multiset is a generalization of set in which elements are allowed to appear more than once. For example, $\{1, 1, 2, 4\}$ is a multiset in which the element 1 appears twice in the multiset. A permutation of a multiset $\mathbf {M}$ is an ordered arrangement of the elements in $\mathbf {M}$. For example, the 12 unique permutations for the multiset $\{1, 1, 2, 4\}$ are:

$$\begin{aligned} 1124, 1214, 2141, 1241, 2411, 4112, 1142, 1412, 4121, 1421, 4211, 2114. \end{aligned}$$

Williams proved that the following simple shift rule exhaustively lists out each of the permutations in a multiset exactly once:

$$\begin{aligned}&{coo\ell }(a_1a_2\cdots a_n) =\left\{ \begin{array}{ll} a_2 a_3 \cdots a_n a_1 &{} \quad \text{ if } \;a_2 a_3 \cdots a_n \\ &{} \quad \text{ is } \text{ strictly } \text{ decreasing; } \\ a_2 a_3 \cdots a_k a_1 a_{k+1} a_{k+2} \cdots a_{n} &{} \quad \text{ if } \; a_1 > a_{k}; \\ a_2 a_3 \cdots a_k a_{k+1} a_1a_{k+2}a_{k+3} \cdots a_{n} &{} \quad \text{ otherwise, } \end{array} \right. \end{aligned}$$

where k is the smallest value such that $a_k < a_{k+1}$. As an illustration, successive applications of this rule for the multiset $\{1, 1, 2, 4\}$ starting with 1124 produce the listing shown earlier in this Section.

A necklace is the lexicographically smallest string in an equivalence class of strings under rotation. Let $f^j(\alpha )$ be the string obtained from j successive applications of the shift rule f starting with $\alpha$. Also, let $\overleftarrow{\alpha }$ denotes the reversal of $\alpha$, that is $\overleftarrow{a_1 a_2 \cdots a_n} = a_n a_{n-1} \cdots a_1$. We now prove Theorem 1 using the cool-lex result by Williams.

Theorem 1

The shift rule f induces a cyclic ordering on ${\varPi }({n})$.

Proof

Let $\alpha = a_1 a_2 \cdots a_n \in {\varPi }({n})$ be a necklace. Thus $a_1 = 1$. By the definition of f, the next $n-1$ strings after $\alpha$ are all possible rotations of $\alpha$. Thus it suffices to show that successive applications of f generate each of the necklaces in ${\varPi }({n})$ exactly once in cyclic order.

Since $a_1 = 1$, $f^{n-1}(\alpha ) = a_n a_1 a_2 \cdots a_{n-1} = a_n 1 a_2 \cdots a_{n-1}$ by the definition of f. Observe that by one more application of f, we have $f(f^{n-1}(\alpha )) = f^{n}(\alpha ) = a_1 \overleftarrow{{coo\ell }(\overleftarrow{a_2 a_3 \cdots a_n})} = 1 \overleftarrow{{coo\ell }(\overleftarrow{a_2 a_3 \cdots a_n})}$. Since the shift rule $coo\ell$ over permutations generates each of the permutations exactly once in cyclic order, successive applications of $\overleftarrow{{coo\ell }(\overleftarrow{a_2 a_3 \cdots a_n})}$ list out each of the permutations of $\{a_2, a_3, \ldots , a_n\}$ exactly once in cyclic order. Thus, successive applications of $f^n$ and $1\overleftarrow{{coo\ell }(\overleftarrow{a_2 a_3 \cdots a_n})}$ on $\alpha$ generate each of the permutations in ${\varPi }({n})$ that starts with the symbol 1 exactly once in cyclic order, that is generating each of the necklaces in ${\varPi }({n})$ exactly once in cyclic order. $\square$

Let U denotes the sequence created by concatenating the first symbol of each permutation in the listing generated by successive applications of f starting with $12\cdots n$. Clearly $|U| = |{\varPi }({n})|$. We then define a function g as follows:

$$\begin{aligned} g(a_1 a_2 \cdots a_n) = a_1 \overleftarrow{coo\ell (\overleftarrow{a_2 a_3 \cdots a_n})}. \end{aligned}$$

From the proof of Theorem 1, the sequence U can be summarized by the following formula:

$$\begin{aligned} U = \alpha _1 \cdot \alpha _2 \cdots \alpha _{{(n-1)!}}, \quad \hbox { where } \; \alpha _1 = 12\cdots n \quad \hbox { and } \quad \alpha _{i+1} = g(\alpha _i). \end{aligned}$$

Corollary 1

Each necklace in ${\varPi }({n})$ appears as a length n substring of U.

We now prove that U is a relaxed shorthand universal cycle for permutations.

Theorem 2

U is a relaxed shorthand universal cycle for ${\varPi }({n})$.

Proof

Since $|U| = |{\varPi }(n)|$, it suffices to show that each of the permutations in ${\varPi }({n})$ appears as a length n substring of U in relaxed shorthand notation.

Let $\beta = b_1 b_2 \cdots b_{n} \in {\varPi }(n)$. If $\beta$ is a necklace, then $\beta$ is a length n substring of U by Corollary 1, that is a length n substring of U in relaxed shorthand notation. Otherwise if $\beta$ is not a necklace, then $\beta$ is a rotation of some necklace $\alpha _i = a_1 a_2 \cdots a_n$, that is $\beta = a_t a_{t+1} \cdots a_n a_1 a_2 \cdots a_{t-1}$ for some $1 < t \le n$. By the definition of U, the next n symbols in U after $\alpha _i$ are $g(\alpha _i ) = \alpha _{i+1} = a_1 a_2 \cdots a_j a_n a_{j+1} \cdots a_{n-1}$ for some $j < n-1$. If $t\le j$, then $\beta$ is a substring of $\alpha _i \cdot a_1 a_2 \cdots a_j$ and U, which is also a length n substring of U in relaxed shorthand notation. Otherwise if $t > j$, then $b_1 b_2 \cdots b_{n-t+j+1} = a_t a_{t+1} \cdots a_n a_1 a_2 \cdots a_j$ and $b_{n-t+j+2} b_{n-t+j+3} \cdots b_{n-1} = a_{j+1} a_{j+2} \cdots a_{t-2}$. Observe that $\gamma = a_t a_{t+1} \cdots a_n a_1 a_2 \cdots a_j a_n a_{j+1} a_{j+2} \cdots a_{t-2} = b_1 b_2 \cdots b_{n-t+j+1} a_n b_{n-t+j+2} b_{n-t+j+3} \cdots b_{n-1}$ is a length n substring of $\alpha _i \alpha _{i+1}$ and U with the symbol $a_n$ appearing twice. Also, $\beta$ can be represented by $\gamma$ in relaxed shorthand notation. Thus, $\beta$ also appears as a length n substring of U in relaxed shorthand notation. Therefore, U is a relaxed shorthand universal cycle for $\varPi (n)$. $\square$

4 Generating our Universal Cycle Efficiently

By Corollary 1, U can be generated by starting with an arbitrary necklace in ${\varPi }(n)$ and repeatedly applying g until it reaches the starting necklace. The function g can be computed in O(n) time. However, g is called only once for every n symbols generated. This leads to an O(1)-amortized time per symbol algorithm to generate U in Algorithm 1. A complete C implementation of the algorithm is given in the Appendix.

Theorem 3

The algorithm UcyclePerm generates the relaxed shorthand universal cycle U for permutations in ${\varPi }({n})$ in O(1)-amortized time per symbol using O(n) space.

Appendix: C code to Generate a Relaxed Shorthand Universal Cycle for Permutations in \({\varPi }({n})\) in O(1)-Amortized Time per Symbol Using O(n) Space