Forbidden Substrings In Circular K-Successions

TL;DR

This paper studies permutations with circular k-successions, counting those avoiding specific substrings, and reveals that the counts depend on the primality and relative primality of n and k.

Contribution

It introduces the concept of circular k-successions and provides formulas for counting permutations avoiding certain substrings based on number-theoretic properties.

Findings

Number of permutations avoiding substrings depends on whether n is prime.

Counts vary with the relative primality of n and k.

Results connect permutation avoidance to number theory.

Abstract

In this note we define circular k-successions in permutations in one-line notation and count permutations that avoid substrings j(j+k) and j(j+k) (mod n). We also count circular permutations that avoid such substrings, and show that for substrings j(j+k) (mod n), the number of permutations depends on whether n is prime, and more generally, on whether n and k are relatively prime.

Tables

Table 1. Table 1: Some values for d n k superscript subscript 𝑑 𝑛 𝑘 d_{n}^{k} .

$n$	$De{r}_n$	$d_n$	$d_n^2$	$d_n^3$	$d_n^4$	$d_n^5$
1	0
2	1	1
3	2	3	4
4	9	11	14	18
5	44	53	64	78	96
6	265	309	362	426	504	600
7	1,854	2,119	2,428	2,790	3,216	3,720
8	14,833	16,687	18,806	21,234	24,024	27,240

Table 2. Table 2: Some values for d ∗ n k subscript superscript superscript 𝑑 ∗ 𝑘 𝑛 {d^{\ast}}^{k}_{n} .

$n$	$d_n^{}{}_{}{}^{\ast }$	$d_{}^{\ast }{}_n{}^2$	$d_{}^{\ast }{}_n{}^3$	$d_{}^{\ast }{}_n{}^4$	$d_{}^{\ast }{}_n{}^5$	$d_{}^{\ast }{}_n{}^6$
1
2	0
3	3	3
4	8	12	16
5	45	55	70	90
6	264	318	384	468	576
7	1,855	2,163	2,534	2,982	3,528	4,200
8	14,832	16,952	19,424	22,320	25,728	29,760

Table 3. Table 3: Some values for c ∗ n k subscript superscript superscript 𝑐 ∗ 𝑘 𝑛 {c^{\ast}}^{k}_{n} .

$n$	$c_{}^{\ast }{}_n{}^{}$	$c_{}^{\ast }{}_n{}^2$	$c_{}^{\ast }{}_n{}^3$	$c_{}^{\ast }{}_n{}^4$	$c_{}^{\ast }{}_n{}^5$	$c_{}^{\ast }{}_n{}^6$
1
2	0
3	1	1
4	2	3	4
5	9	11	14	18
6	44	53	64	78	96
7	265	309	362	426	504	600
8	1,854	2,119	2,428	2,790	3,216	3,720

Table 4. Table 4: Some values for D n k subscript superscript 𝐷 𝑘 𝑛 D^{k}_{n} .

	$k=1$	$k=2$	$k=3$	$k=4$	$k=5$	$k=6$
$n=2$	0
$n=3$	3	3
$n=4$	8	8	8
$n=5$	45	45	45	45
$n=6$	264	270	240	270	264
$n=7$	1,855	1,855	1,855	1,855	1,855	1,855

Table 5. Table 5: Some values for D ∗ n k subscript superscript superscript 𝐷 ∗ 𝑘 𝑛 {D^{\ast}}^{k}_{n} .

	$k=1$	$k=2$	$k=3$	$k=4$	$k=5$	$k=6$
$n=2$	0
$n=3$	3	3
$n=4$	4	8	4
$n=5$	40	40	40	40
$n=6$	216	234	192	234	216
$n=7$	1,603	1,603	1,603	1,603	1,603	1,603

Table 6. Table 6: Some values for C ∗ n k subscript superscript superscript 𝐶 ∗ 𝑘 𝑛 {C^{\ast}}^{k}_{n} .

	$k=1$	$k=2$	$k=3$	$k=4$	$k=5$	$k=6$
$n=2$	0
$n=3$	1	1
$n=4$	1	2	1
$n=5$	8	8	8	8
$n=6$	36	39	32	39	36
$n=7$	229	229	229	229	229	229

Equations

$$d_n^k=\sum _{j=0}^{n-k}(-1{)}^j(\genfrac{}{}{0ex}{}{n-k}{j})(n-j!).$$

$$D_n^k=\sum _{j=0}^{n-1}(-1{)}^j(\genfrac{}{}{0ex}{}{n}{j})(n-j)!.$$

$$c=\sum _{j=0}^{n-k}(-1{)}^j(\genfrac{}{}{0ex}{}{n-k}{j})(n-j-1)!$$

$${c^{\ast }}_n^k=\sum _{j=0}^{n-k}(-1{)}^j(\genfrac{}{}{0ex}{}{n-k}{j})(n-j-1)!.$$

$${c_n}^{\ast }=Der(n-1),$$

$$Der(n)=\sum _{j=0}^n(-1{)}^j(\genfrac{}{}{0ex}{}{n}{j})(n-j)!$$

$${D^{\ast }}_n=n\left[\sum _{j=0}^{n-1}(-1{)}^j(\genfrac{}{}{0ex}{}{n}{j})(n-j-1)!+(-1{)}^n\right].$$

$$C=\sum _{j=0}^{n-1}(-1{)}^j(\genfrac{}{}{0ex}{}{n}{j})(n-j-1)!+(-1{)}^n$$

$${C^{\ast }}_n^k=\sum _{j=0}^{n-1}(-1{)}^j(\genfrac{}{}{0ex}{}{n}{j})(n-j-1)!+(-1{)}^n.$$

Full text

Keywords: Circular permutations, circular successions, circular$k$-successions, $k$-shifts, $k$-successions, derangements, forbidden substrings, bijections.

1.   Introduction and Previous Results

In [3] we counted the number of permutations according to $k$-shifts, where we defined $\{d^{k}_{n}\}$ as the set of permutations on $[n]$ that for fixed $k$, $k<n$, avoid substrings $j(j+k)$, $1\leq j\leq n-k$. The number of such permutations, $d^{k}_{n}$, turned out to be

[TABLE]

The forbidden substrings in these permutations can be pictured as a diagonal running $k$ places to the right of the main diagonal of an $n\times n$ chessboard (hence the term “$k$-shifts”). Note that we will also refer to these forbidden substrings that are $k$ spacings apart as “$k$-successions”; therefore in [3] we counted permutations with no $k$-successions for $k>1$ (the case $k=1$ was discussed in [2]). Note that there are some references that not only count permutations with no successions but also count permutations with$i=1,2,\ldots,n-1$ successions (the so-called “succession numbers”) but they do it only for the case $k=1$, ie. for forbidden substrings that are only one spacing apart, $j(j+1)$, so some care should be taken (see [1], for example).

Table 1 in the Appendix provides some $d^{k}_{n}$ values. For example, for $n=4$, $k=2$, the forbidden substrings that are 2 spacings apart are $\{13,24\}$, and $d^{2}_{4}=14$ since there are 14 permutations in $S_{4}$ that avoid such substrings or 2-successions.

The sequence $\langle d^{k}_{n}\rangle$ is available in OEIS. For example, for $k=4$, the sequence is now A277563 [4], and for $k=1$ it is A000255 [5]. Note that for $k=0$, $d^{k}_{n}=Der(n)$, the $n$th derangement number, which is A000166 in OEIS [9].

In [3] we also counted the number of permutations according to $k$-shifts $(\text{mod }n)$, where we defined $\{D^{k}_{n}\}$ as the set of permutations on $[n]$ that for fixed $k$, $k<n$, avoid substrings $j(j+k)$ for $1\leq j\leq n-k$, and avoid substrings $j(j+k)\ (\text{mod }n)$ for $n-k<j\leq n$. Note that we can summarize in the single definition “avoid substrings $j(j+k)\ (\text{mod }n)$ for all $j$, $1\leq j\leq n$”  if we agree to write $n$ instead of 0 when doing addition $(\text{mod }n)$.

The forbidden substrings in this case are easily seen along an $n\times n$ chessboard, where for $j>n-k$, the forbidden positions start again from the first column along a diagonal $n-k$ places below the main diagonal, as in Figure 1 below.

Figure 1 shows forbidden positions on a $4\times 4$ chessboard that correspond to forbidden substrings of permutations in $\{D^{3}_{4}\}$. These forbidden substrings are $\{14;21,32,43\}$. The forbidden substrings below the diagonal are separated by a semicolon; these are the forbidden substrings $j(j+k)\ (\text{mod }n)$ for $n-k<j\leq n$. Note that while there are only $n-k$ forbidden substrings in $\{d^{k}_{n}\}$, there are $n$ forbidden substrings in $\{D^{k}_{n}\}$.

The number of permutations in $\{D^{k}_{n}\}$ is given by

[TABLE]

Furthermore, there are the same number of permutations in $\{D^{k}_{n}\}$ as in $\{D_{n}\}$ whenever $n$ and $k$ are relatively prime ($D_{n}=D^{1}_{n}$; see [3]). Table 4 in the Appendix provides a table for some $D^{k}_{n}$ values. For example, for $n=4$, $k=3$, we have seen that forbidden substrings in $\{D^{3}_{4}\}$ are $\{14;21,32,43\}$ and $D^{3}_{4}=4$ since there are 4 permutations in $S_{4}$ that avoid such substrings or 3-successions. Furthermore, since $(4,3)=1$, there is the same number of permutations in $\{D^{3}_{4}\}$ as in $\{D_{4}\}$, which are permutations in $S_{4}$ that avoid successions $\{12,23,34;41\}$. The sequence $\langle D_{n}\rangle$ is A000240 in OEIS [6].

In this note we will discuss permutations in $\{d^{k}_{n}\}$ and $\{D^{k}_{n}\}$ with no circular $k$-successions, as described in the following definition.

Definition 1.1**.**

We define a $circular\ k\text{-}succession$ in a permutation $p$ on $[n]$ as either a pair $p(i)$, $p(i+1)$ if $p(i+1)\equiv p(i)+k\ (\text{mod }n)$, or as the pair $p(n)$, $p(1)$ if $p(1)\equiv p(n)+k\ (\text{mod }n)$. We will denote as $\{{d^{\ast}}^{k}_{n}\}$ the set of permutations in $\{d^{k}_{n}\}$ that have no circular $k$-successions, and similarly, as $\{{D^{\ast}}^{k}_{n}\}$ the set of permutations in $\{D^{k}_{n}\}$ that have no circular $k$-successions. These permutations are in one-line notation.

Hence permutations in $\{{d^{\ast}}^{k}_{n}\}$ avoid substrings $j(j+k)$, $1\leq j\leq n-k$, and permutations in $\{{D^{\ast}}^{k}_{n}\}$ avoid substrings $j(j+k)\ (\text{mod }n)$ for all $j$, $1\leq j\leq n$ (as in $\{d^{k}_{n}\}$ and $\{D^{k}_{n}\}$, respectively), but in $\{{d^{\ast}}^{k}_{n}\}$ and $\{{D^{\ast}}^{k}_{n}\}$ there is the additional restriction of the circular $k$-successions.

For example, in $\{D^{3}_{4}\}$, we saw above that the forbidden substrings are$\{14;21,32,43\}$, hence the permutation 2134 has a circular 3-succession 21 and the permutation 2413 has a circular 3-succession 32. Hence such permutations won’t be allowed in $\{{D^{\ast}}^{3}_{4}\}$ (but 2413 is allowed in $\{D^{3}_{4}\}$). Note as mentioned above that the forbidden substrings in $\{{D^{\ast}}^{k}_{n}\}$ are the same as those in $\{D^{k}_{n}\}$ and the same is true for $\{{d^{\ast}}^{k}_{n}\}$ and $\{d^{k}_{n}\}$, with differences in cardinalities being accounted by circular $k$-successions, as will be shown below.

Note that in Definition 1.1 we don’t really need the definition $(\text{mod }n)$ in $\{{d^{\ast}}^{k}_{n}\}$ since forbidden positions in those permutations stay above the main diagonal of an $n\times n$ chessboard.

2.   Main Lemmas and Propositions

In [2] we divided permutations according to their starting digit and called these groups classes. We showed that permutations in $\{D_{n}\}$ are equidistributed, meaning that all classes are equinumerous. This is also true for deranged permutations (see [2]).

Now we note that in general $\{{d^{\ast}}^{k}_{n}\}$, which forbids permutations with circular $k$-successions, will have less permutations than $\{d^{k}_{n}\}$. However, if we look at individual classes starting with digits $1,2,\ldots,n$, we see that there will be classes in $\{d^{k}_{n}\}$ and $\{{d^{\ast}}^{k}_{n}\}$ that have the same number of elements, hence they are the same.

Indeed, since forbidden substrings in $\{d^{k}_{n}\}$ run strictly over the main diagonal, there will be substrings $j(j+k)$ where $(j+k)\neq i$, $i=1,2,\ldots,k$, hence $\{{d^{\ast}}^{k}_{n}\}$ will not have circular $k$-successions starting with $i=1,2,\ldots,k$. Hence classes in $\{d^{k}_{n}\}$ and $\{{d^{\ast}}^{k}_{n}\}$ starting with these digits will be the same, and classes starting with $i=k+1,\ldots,n$ will be smaller in $\{{d^{\ast}}^{k}_{n}\}$ than in $\{d^{k}_{n}\}$. Hence a total of $k$ classes will be the same in $\{{d^{\ast}}^{k}_{n}\}$ and in $\{d^{k}_{n}\}$, and $n-k$ classes will be smaller in $\{{d^{\ast}}^{k}_{n}\}$ than in $\{d^{k}_{n}\}$, where $n-k$ is the number of forbidden substrings in $\{d^{k}_{n}\}$ and $\{{d^{\ast}}^{k}_{n}\}$.

For example, for $n=4$, $k=3$, the forbidden substring in $\{d^{3}_{4}\}$ is $\{14\}$, so there won’t be permutations in $\{{d^{\ast}}^{3}_{4}\}$ ending in 1 and starting with 4, since they would represent circular 3-successions, which are forbidden. Hence the class of permutations starting with 4 is smaller in $\{{d^{\ast}}^{3}_{4}\}$ than in $\{d^{3}_{4}\}$ and the other classes starting with $i=1,2,3$ are equal in both cases. In fact, the cardinalities of the four classes in $\{{d^{\ast}}^{3}_{4}\}$ are $\{4,4,4,4\}$, while the cardinalities of the four classes in $\{d^{3}_{4}\}$ are $\{4,4,4,6\}$, the difference being accounted by the permutations 4231 and 4321 which are allowed in $\{d^{3}_{4}\}$ but not in $\{{d^{\ast}}^{3}_{4}\}$.

Similarly, for $n=6$, $k=2$, forbidden substrings in $\{d^{2}_{6}\}$ are $\{13,24,35,46\}$, so there will be forbidden circular 2-successions in $\{{d^{\ast}}^{2}_{6}\}$ starting with 3, 4, 5, 6 but not with 1 and 2. Therefore, the classes in $\{d^{2}_{6}\}$ and in $\{{d^{\ast}}^{2}_{6}\}$ starting with 1 and 2 will be the same, and the other classes will be smaller in $\{{d^{\ast}}^{2}_{6}\}$ than in $\{d^{2}_{6}\}$. In fact, the cardinalities of the six classes in $\{d^{2}_{6}\}$ are $\{53,53,64,64,64,64\}$, for a total of 362 permutations, while the cardinality of the classes in $\{{d^{\ast}}^{2}_{6}\}$ is 53 in all six cases, for a total of 318. Hence there are 318 permutations in $S_{6}$ that have no circular 2-successions, ie. permutations that avoid substrings such as 13 in 213456 or in 345621.

From these examples one might suspect that permutations in $\{{d^{\ast}}^{k}_{n}\}$ areequidistributed (equinumerous), a conjecture which is confirmed in the next section.

2.1 Results for permutations in $\boldsymbol{\{{d^{\ast}}^{k}_{n}\}}$

Proposition 2.1**.**

For fixed $k$, $1\leq k\leq n-1$, if ${d^{\ast}}^{k}_{n}$ denotes the number of permutations on $[n]$ that avoid circular $k$-successions, then ${d^{\ast}}^{k}_{n}=nd^{k-1}_{n-1}$.

Proof. Consider a class in $\{{d^{\ast}}^{k}_{n}\}$ where both $\{{d^{\ast}}^{k}_{n}\}$ and $\{d^{k}_{n}\}$ have the same number of elements (there is at least one such class for $1\leq k\leq n-1$), and denote the cardinality of such a class by $c$. Since a circular permutation is an $n$-to-1 mapping with respect to a linear one, we have that ${d^{\ast}}^{k}_{n}=n\cdot c$. By inclusion-exclusion, it is straightforward to show that

[TABLE]

since the combinatorial term counts the number of ways to get substrings of length $j$ while the term $(n-j-1)!$ counts the $(n-j)!/(n-j)$ circular permutations of forbidden substrings of length $j$ and the remaining elements. Hence, by Equation (1.1), we see that the right-hand side of Equation (2.1) is just $d^{k-1}_{n-1}$ and the proof follows. $\square$

For example, in $\{d^{2}_{4}\}$ we see that the forbidden substrings are $\{13,24\}$. Recall from [3] that we define a minimal forbidden substring as two consecutive elements $ik$. We assign to this minimal substring a length equal to one. Recall also that a forbidden substring of length $j$ can be considered as either a single substring of length $j$ or as $j$ overlapping substrings of length 1.

Hence for the term $j=0$ in Equation 2.1 we see that we can choose no forbidden substrings in $\binom{2}{0}$ ways and we can permute them circularly with the remaining elements 1,2,3,4 in $4!/4$ ways. For the term $j=1$ we can choose 1 forbidden substring in $\binom{2}{1}$ ways and we can permute it circularly with the remaining elements in $3!/3$ ways (for example, choose the substring 13 and permute the blocks 13, 2, 4 circularly in $3!/3$ ways). For the term $j=2$ we can choose 2 forbidden substrings in $\binom{2}{2}$ ways and we can permute them circularly with the remaining elements in $2!/2$ ways (that is, permute the blocks 13, 24 circularly in $1!$ way). Then we have that ${d^{\ast}}^{k}_{n}=n\cdot c=4\cdot 3$, so there are 12 permutations in $S_{4}$ that avoid substrings $\{13,24\}$ in circular 2-successions (for example, permutations such as 1342 and 3421 are not allowed).

Note that even though permutations in $\{d^{k}_{n}\}$ are not equidistributed, we gain this property when we forbid circular $k$-successions in $\{d^{k}_{n}\}$, as recorded in the following corollary, which follows from Proposition 2.1, ${d^{\ast}}^{k}_{n}=nd^{k-1}_{n-1}$.

Corollary 2.2**.**

The classes in $\{{d^{\ast}}^{k}_{n}\}$ are equidistributed (equinumerous) and the cardinality of each class is given by $d^{k-1}_{n-1}$.

Note that the property of classes being equidistributed is not shared by all types of permutations with forbidden substrings. For example, $\{d_{n}\}$ and $\{d^{k}_{n}\}$ don’t have this property, while $\{D_{n}\}$ and $\{D^{k}_{n}\}$ do.

Now we define ${c^{\ast}}^{k}_{n}$ as the number of circular permutations on $[n]$ (in cycle notation) that avoid substrings $j(j+k)$, $1\leq j\leq n-k$, ie. the same forbidden substrings as in $\{d^{k}_{n}\}$ and $\{{d^{\ast}}^{k}_{n}\}$. We have the following corollary that counts such permutations.

Corollary 2.3**.**

For fixed $k$, $1\leq k\leq n-1$, the number of circular permutations ${c^{\ast}}^{k}_{n}$ on $[n]$ that avoid substrings $j(j+k)$, $1\leq j\leq n-k$, is given by

[TABLE]

Proof. This is the number $c$ from the proof of Proposition 2.1. Hence we may write ${d^{\ast}}^{k}_{n}=n{c^{\ast}}^{k}_{n}$ , where ${c^{\ast}}^{k}_{n}=d^{k-1}_{n-1}$. $\square$

For example, ${c^{\ast}}^{2}_{5}=d^{1}_{4}=11$, so there are 11 circular permutations in $\{{c^{\ast}}^{2}_{5}\}$ that avoid substrings $\{13,24,35\}$. Furthermore, there are $5\cdot 11=55$ permutations in $\{{d^{\ast}}^{2}_{5}\}$ that avoid such substrings in circular 2-successions.

Similarly, ${c^{\ast}}^{2}_{4}=d^{1}_{3}=3$, so there are 3 circular permutations that avoidsubstrings $\{13,24\}$. These are the permutations $\{(1234),(1423),(1432)\}$, and since $n=4$, to each of these circular permutations correspond 4 permutations in $\{{d^{\ast}}^{2}_{4}\}$ with no circular 2-succession (for example the ones for (1234) are$\{1234,2341,3412,4123\}$).

Corollary 2.4**.**

The number of circular permutations on $[n]$ that avoid substrings $j(j+1)$, $1\leq j\leq n-1$, (ie. $k=1$) is given by

[TABLE]

where $Der(n)$ is the $n$th derangement number.

Proof. Since we have that

[TABLE]

and since Equation 2.2 holds for $k=1$, the result is immediate $({c^{\ast}}^{1}_{n}={c^{\ast}}_{n})$. $\square$

For example, for $n=3$, $k=1$, the number of circular permutations that avoid substrings $\{12,23\}$ is $Der(2)=1$ and this circular permutation is $(132)$, which corresponds to the 3 permutations $\{132,213,321\}$ in $\{{d^{\ast}}_{3}\}$.

Tables 1 - 3 in the Appendix show some values for $d^{k}_{n}$, ${d^{\ast}}^{k}_{n}$, and ${c^{\ast}}^{k}_{n}$.

2.2 Results for permutations in $\boldsymbol{\{{D^{\ast}}^{k}_{n}\}}$

As in the case of $D^{k}_{n}$ , the numbers ${D^{\ast}}^{k}_{n}$ are more difficult to get. As opposed to permutations in $\{{d^{\ast}}^{k}_{n}\}$, permutations in $\{{D^{\ast}}^{k}_{n}\}$ avoid circular $k$-successions starting with all digits since a total of $n$ substrings are forbidden (as opposed to $n-k$ forbidden substrings in $\{{d^{\ast}}^{k}_{n}\}$).

We first count the number of permutations in $\{D^{\ast}_{n}\}$, (ie. $k=1$). We have the following proposition.

Proposition 2.5**.**

The number of permutations in $\{{D^{\ast}}_{n}\}$ is given by

[TABLE]

Proof. As in the proof of Proposition 2.1, we first obtain the number of circular permutations without forbidden substrings. It is easy to see that the number of such permutations is given by

[TABLE]

since the combinatorial term counts the number of ways to get forbiddensubstrings while the term $(n-j-1)!$ counts the $(n-j)!/(n-j)$ circular permutations of forbidden substrings and the remaining elements. The result is then obtained by inclusion-exclusion, where for the term $j=n$, we note that the only way to get an $n$-element substring is by the circular permutation $(12\ldots n)$. $\square$

For example, the forbidden substrings in $\{{D^{\ast}}_{4}\}$ are $\{12,23,34;41\}$. It is easy to count, for instance, that there are $\binom{4}{2}$ forbidden substrings of length $j=2$ and $2!/2$ circular permutations of these substrings and the remaining elements. For example, a substring of length 2 is given by 123 and we count$2!/2=1!$ circular permutation of the blocks 123, 4. Another substring of length 2 (alternatively, two substrings of length 1) is given by 12 34 and we also count $2!/2=1!$ circular permutation of these two blocks. Furthermore, the only way to get $j=4$ forbidden substrings is by the circular permutation $(1234)$, which produces the forbidden substrings $\{12,23,34;41\}$. Hence $C=1$ by Equation 2.6 and by Equation 2.5 we have that ${D^{\ast}}_{4}=4$, which counts the permutations $\{1432,2143,3214,4321\}$ in $S_{4}$ that have no circular succession, (ie. no circular 1-succession) and avoid substrings $\{12,23,34;41\}$.

If we now move on to permutations without $k$-successions that avoid substrings $j(j+k)\ (\text{mod }n)$ for $1\leq j\leq n$, we see that as in the case of $\{D^{k}_{n}\}$, the number of permutations in $\{{D^{\ast}}^{k}_{n}\}$ depends on whether $n$ is prime, and more generally, on whether $n$ and $k$ are relatively prime. Since the proof follows closely the one for $\{D^{k}_{n}\}$, we refer the reader to [3] for the details.

We will only record the main counting result in the proposition below.

Proposition 2.6**.**

The number of permutations in $\{{D^{\ast}}^{k}_{n}\}$ with $n$ relative prime to $k$, $n\geq 3$, $k<n$, is the same as the number of permutations in $\{{D^{\ast}}_{n}\}$.

The key result in the proof of the proposition is that, if $(n,k)=1$, there are forbidden substrings of all lengths in $\{D^{k}_{n}\}$ (see [3]). But forbidden substrings are the same in $\{D^{k}_{n}\}$ and in $\{{D^{\ast}}^{k}_{n}\}$, so in this case there will also exist forbidden substrings of all lengths in $\{{D^{\ast}}^{k}_{n}\}$.

The proposition implies the following corollary:

Corollary 2.7**.**

The number of permutations in $\{{D^{\ast}}^{k}_{p}\}$, $k=1,2,\ldots,p-1$, is the same as the number of permutations in $\{{D^{\ast}}_{p}\}$, $p$ prime, $p\geq 3$.

Proof. $(p,k)=1$, $k=1,2,\ldots,p-1$. $\square$

Note that Proposition 2.6 is not true if $n$ is not relative prime to $k$, for example in $\{{D^{\ast}}^{2}_{4}\}$. In this case the forbidden substrings are $\{13,24;31,42\}$, and we can’t get substrings of lengths 3 or 4. On the other hand, since $(4,3)=1$, we see that forbidden substrings in $\{{D^{\ast}}^{3}_{4}\}$ are $\{14;21,32,43\}$, and we can get substrings of all lengths. For example, 2143 is a substring of length 3 (recall that the unit forbidden substring $ik$ has length one, hence the length of forbidden substrings will be one less than the number of elements), and a substring of length 4 (alternatively, 4 substrings of length 1) is given by the circular permutation (1432). We then have that ${D^{\ast}}_{4}=4$ by Equation 2.5 and hence ${D^{\ast}}^{3}_{4}=4$ by Proposition 2.6.

As in the case of $\{{d^{\ast}}^{k}_{n}\}$, permutations in $\{{D^{\ast}}^{k}_{n}\}$ are equidistributed, as established by the following corollary:

Corollary 2.8**.**

The classes in $\{{D^{\ast}}^{k}_{n}\}$ are equidistributed (equinumerous) and the cardinality of each class is given by the term $C$ in Equation 2.6.

Proof. Equation 2.5 in Proposition 2.5. $\square$

We now define ${C^{\ast}}^{k}_{n}$ as the number of circular permutations on $[n]$ (in cycle notation) that avoid substrings $j(j+k)\ (\text{mod }n)$ for $1\leq j\leq n$, ie. the same substrings as in $\{D^{k}_{n}\}$ and $\{{D^{\ast}}^{k}_{n}\}$. We have the following corollary that counts such permutations for $n$ and $k$ relatively prime.

Corollary 2.9**.**

For $(n,k)=1$, the number of circular permutations ${C^{\ast}}^{k}_{n}$ on $[n]$ that avoid substrings $j(j+k)\ (\text{mod }n)$ for $1\leq j\leq n$, is given by

[TABLE]

Proof. This is the number $C$ from the proof of Proposition 2.5. Since forbidden substrings in $\{{C^{\ast}}^{k}_{n}\}$ are the same as those in $\{{D^{\ast}}^{k}_{n}\}$ and since$(n,k)=1$, there are forbidden substrings of all lengths $j=0,1,2,\ldots,n-1,n$, so the sum in Equation 2.7 applies. $\square$

For example, since $(4,3)=1$, we use Equation 2.7 to compute ${C^{\ast}}^{3}_{4}=1$, so there is only one circular permutation that avoids substrings $\{14;21,32,43\}$. This circular permutation is $(1234)$, which corresponds to the four permutations $\{1234,2341,3412,4123\}$ in $\{{D^{\ast}}^{3}_{4}\}$ with no circular 3-succession. Since $(4,1)=1$, we also have ${C^{\ast}}^{1}_{4}={C^{\ast}}_{4}=1$ by Equation 2.7, and the circular permutation that avoids substrings $\{12,23,34;41\}$ is $(1432)$, which corresponds to thepermutations $\{1432,2143,3214,4321\}$ in $\{{D^{\ast}}_{4}\}$, as seen above.

We have that the gcd of $n$ and $k$ is very important to determine the number of permutations in $\{{C^{\ast}}^{k}_{n}\}$. In fact, if for the same $n$ we have that $(n,k_{1})=(n,k_{2})$, then $\{{C^{\ast}}^{k_{1}}_{n}\}$ and $\{{C^{\ast}}^{k_{2}}_{n}\}$ will have the same number of permutations. The same is true for $\{{D^{\ast}}^{k}_{n}\}$.

Note that usually ${C^{\ast}}^{k}_{n}\neq{c^{\ast}}^{k}_{n}$ except for the case $n=3$ (and trivially for $n=2$). Indeed, we have that ${C^{\ast}}_{3}={c^{\ast}}_{3}=1$ since both numbers count the permutation (132), and ${C^{\ast}}^{2}_{3}={c^{\ast}}^{2}_{3}=1$ since both numbers count the permutation $(123)$. This in turn implies that ${D^{\ast}}_{3}={d^{\ast}}_{3}=3$ and ${D^{\ast}}^{2}_{3}={d^{\ast}}^{2}_{3}=3$.

As a final remark for the case $k=1$ in circular permutations, recall that${c^{\ast}}^{1}_{n}={c^{\ast}}_{n}$, and this counts the number of circular permutations that avoid substrings $j(j+1)$, $1\leq j\leq n-1$; ie. substrings ${12,23,\ldots,(n-1)n}$. By Corollary 2.4, ${c_{n}}^{\ast}=Der(n-1)$, so for example, for $n=4$, we have that ${c^{\ast}}_{4}=Der(3)=2$, and the 2 permutations that avoid such substrings are $\{(1324),(1432)\}$.

On the other hand, ${C^{\ast}}^{1}_{n}={C^{\ast}}_{n}$, and this counts the number of permutations that avoid substrings $j(j+1)\ (\text{mod }n)$ for $1\leq j\leq n$; ie. substrings$\{12,23,\ldots,(n-1)n,n1\}$. By Corollary 2.9, since $(4,1)=1$, we have seen that ${C^{\ast}}^{1}_{4}=1$ and the single permutation that avoids such substrings is $(1432)$ (the permutation $(1324)$ from $\{{c^{\ast}}_{4}\}$ is excluded since it has the succession 41, which is forbidden in $\{{C^{\ast}}_{4}\}$).

Note that while in some references such as [1] ${C^{\ast}}_{n}$ is referred to as “the number of circular permutations without a succession”, we also consider circularpermutations without a succession in $\{{c^{\ast}}_{n}\}$ on the smaller subset of forbidden substrings $\{12,23,\ldots,(n-1)n\}$. Furthermore, the same reference counts circular permutations with $i=1,2,\ldots,n-1$ successions (the so-called “circularsuccession numbers”) but it does it only for forbidden substrings that are one spacing apart, $j(j+k)$, $k=1$.

For $k>1$, this note generalizes the enumeration of circular permutations without $k$-successions (or $k$-shifts) for both kinds of forbidden substrings (above the diagonal of an $n\times n$ chessboard in $\{{c^{\ast}}^{k}_{n}\}$ and both above and below the diagonal, ie. $(\text{mod }n)$ in $\{{C^{\ast}}^{k}_{n}\}$). It also enumerates the correspondingpermutations in one-line notation in $\{{d^{\ast}}^{k}_{n}\}$ and $\{{D^{\ast}}^{k}_{n}\}$.

For further references, the sequence $\langle{C^{\ast}}_{n}\rangle$ is A000757 in OEIS [7], $\langle{D^{\ast}}_{n}\rangle$ is A167760 [8], $\langle{d^{\ast}}_{n}\rangle$ is A000240 [6], and $\langle{c^{\ast}}_{n}\rangle$ can be looked up in the derangement numbers A000166 [9] due to Corollary 2.4. Note that A000240 not only counts the number of permutations of $[n]$ having no circular succession, ${d^{\ast}}_{n}$, but also the number of permutations $D_{n}$ on $[n]$ having no substring in$\{12,23,\ldots,(n-1)n,n1\}$, as well as the number of permutations of $[n]$ having exactly one fixed point (see [2]).

Tables 4 - 6 in the Appendix show some values for $D^{k}_{n}$, ${D^{\ast}}^{k}_{n}$, and ${C^{\ast}}^{k}_{n}$.