The Lehmer factorial norm on Sn

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The Lehmer factorial norm on Sn

Paweł ZawiślakDepartment of Mathematics and Mathematical Economics

SGH Warsaw School of EconomicsAl. Niepodległości 162, 02-554 Warszawa, Poland

E-mail: [email protected]

November 9, 2021

Abstract

We introduce a new family of norms on the permutation groups Sn. We

examine their properties.

1 Introduction

Metrics on the permutation groups Sn were considered in many different contexts.

On one side, permutations can be used used as rankings, therefore some metrics

on permutations originate from attempts of comparing rankings. Many well known

measures of similarity between rankings lead to definition of metric on Sn. The

most popular measures of similarity are Kendall’s τ ([K]) and Spearman’s ρ ([S]).

These two measures leads to Kendall’s distance and Spearman’s distance. Together

with Spearman footrule (also known as Manhattan distance) and Hamming distance,

these four metrics are the most popular metrics on Sn used in the statistics ([DG]

and [DH]).

2020 Mathematics Subject Classification: 05A05, 62H20, 54E35, 20B99

Key words and phrases: Lehmer code, Lehmer factorial norm, permutation

1

http://arxiv.org/abs/2111.03951v1

2 P. Zawiślak

The generalisations of these metrics in many different contexts we considered for

example in [KW], [LH], [LZH], [KV], [QDRL], [LY1], [LY2], [WSSC], [FSS] and [PP].

On the other side, the study of statistical properties of natural valued functions on

Sn has almost two hundred years of history, started with [R] and [M], and continued

by many authors (see for example [FZ], [SS] and [CSZ]).

On the third side, natural inclusions of Sk in Sl (for k < l) lead to a limit object

S∞. Metrics on some limit objects related to groups were explored for example in

[C], [TW], [W] and [GMZ].

In this paper we present a slightly different approach, which can be considered

as transversal to the previous three. The demand for the metric satisfying conditions

(i)-(vii) of Theorem 3.6 comes from the analysis of the votings’ networks. Presicely,

all most popular metrics on Sn do not differentiate between the change on first two

positions of the ranking and the change on last two positions. The metric coming from

the norm presented in this paper do – see Theorem 3.6 (iii). Additionally, Theorems

4.5 and 4.6 describe distributions of the new norm (we call it the Lehmer norm) on

all permutation groups Sn as well as its distribution on S∞.

We do not consider many others research contexts of permutations metrics. For

more of these contexts see for example [DH].

This paper is organised as follows. Section 2 contains the basic definitions and

notation. In Section 3 we define the main object of this article - the Lehmer factorial

norm. The definition bases on the notion on the Lehmer code. The properties of

the Lehmer code are described in Lemmas 3.1 and 3.3 as well as in Corollary 3.2.

Theorem 3.6 contains the basic attributes of the Lehmer norm. In Section 4 we

focus on the distribution of the Lehmer norm. This distribution is fully described in

Theorems 4.5 and 4.6 together with Lemma 4.4.

2 Basic definitions and notation

In this section we recall some basic definitions used in this paper as well as we set

some notation.

In this article N denotes the set of all natural numbers, starting at 0, whereas

N+ – the set of all positive natural numbers. For n ∈ N+ by [n] we denote the set

The Lehmer factorial norm on Sn 3

{1,2, . . . , n} and by Sn – the group of all permutations of [n]. S∞ stands for the

group of all permutations of N+ with a finite support.

A permutation σ ∈ Sn is denoted by

σ = (σ(1), σ(2), . . . , σ(n))In particular en = (1,2, . . . , n) denotes the identity permutation.

By σ−1 we denote the inverse permutation to σ, by στ – the composition of σ

and τ , defined by (στ)(i) = σ(τ(i)) for i = 1,2, . . . , n, whereas σ stands for the

permutation reverse to σ, given by σ(i) = σ(n + 1 − i) for i = 1,2, . . . , n.

For s = 1,2, . . . , n − 1 let

σs = (1,2, . . . , s − 1, s + 1, s, s + 2, . . . , n)(so σs is the adjacent transposition – (s, s + 1) in the cycle notation).

Definition 2.1. For a permutation σ ∈ Sn its Lehmer code lc(σ) (see [G]) is defined

by

lc(σ) = [c1(σ), c2(σ), . . . , cn(σ)]where the numbers ci(σ) (for i = 1,2, . . . , n) are given by

ci(σ) = ∣{j ∈ [n] ∶ j > i and σ(j) < σ(i)}∣The Lehmer code of σ coincides with the factorial number system representa-

tion of its position in the list of permutations of [n] in the lexicographical order

(numbering the positions starting from 0) – compare [G] to [L1] and [L2].

The Lehmer codes of permutations σ ∈ S3 are presented in Table 1.

3 The Lehmer factorial norm

In this section we define the Lehmer factorial norm on the group Sn. We also examine

its basic features.

We start with establishing some basic properties of the Lehmer code. To do this,

we need some technical notation. For a permutation σ ∈ Sn and for i = 1,2, . . . , n let

C(σ)i = {j ∈ [n] ∶ j > i and σ(j) < σ(i)} and A(σ)i = [i] ∪C(σ)i

4 P. Zawiślak

Note, that if we denote the cardinality of X by ∣X ∣, then ∣C(σ)i∣ = ci(σ) and ∣A(σ)i∣ =i + ci(σ).Lemma 3.1. For all permutations σ, τ ∈ Sn and for all i = 1,2, . . . , n the following

hold:

(i) σ(i) ≤ i + ci(σ),(ii) ci(στ) ≤ ci(τ) + cτ(i)(σ),(iii) σ−1 determines the bijection between A(σ−1)σ(i) and A(σ)i. In particular

i + ci(σ) = σ(i) + cσ(i)(σ−1)Proof. (i) Note that ∣{j ∈ [n] ∶ j > i}∣ = n − i, so

∣{j ∈ [n] ∶ j > i and σ(j) > σ(i)}∣ = n − i − ci(σ)On the other hand

∣{σ(j) ∶ j ∈ [n] and σ(j) > σ(i)}∣ = n − σ(i)And since

σ[{j ∈ [n] ∶ j > i and σ(j) > σ(i)}] ⊆ {σ(j) ∶ j ∈ [n] and σ(j) > σ(i)}it follows that n − i − ci(σ) ≤ n − σ(i).

(ii) Choose k ∈ C(στ)i. If τ(k) < τ(i), then k ∈ C(τ)i. Otherwise τ(k) > τ(i) and

σ(τ(k)) < σ(τ(i)), therefore τ(k) ∈ C(σ)τ(i).(iii) Choose k ∈ A(σ−1)σ(i) and let l = σ−1(k). We will show that l ∈ A(σ)i. There

are two possible cases.

(a) k ≤ σ(i): If l ≤ i, then l ∈ [i] ⊆ A(σ)i. Otherwise l > i, and since σ(l) = k ≤ σ(i),it follows that σ(l) < σ(i) hence l ∈ C(σ)i ⊆ A(σ)i.

(b) k > σ(i): Therefore k ∈ C(σ−1)σ(i), so l = σ−1(k) < σ−1(σ(i)) = i and thus

l ∈ [i] ⊆ A(σ)i.


We have shown that σ−1 [A(σ−1)σ(i)] ⊆ A(σ)i. Replacing σ with σ−1 leads to the

second inclusion, which completes the proof.

As an obvious conclusion of Lemma 3.1 (iii) we get:

Corollary 3.2. Elements of the Lehmer code of the inverse permutation to σ are

given by

ci(σ−1) = cσ−1(i)(σ) + σ−1(i) − i

for i = 1,2, . . . , n.

In the next lemma we describe how the Lehmer code changes when a permutation

is multiplied by an adjacent transposition.

Lemma 3.3. (i) ci(σs) = δis (the Kronecker delta) for i = 1,2, . . . , n and s =1,2, . . . , n − 1.

(ii) Let τ = σσs. If σ(s) < σ(s + 1), then

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩ci(τ) = ci(σ) for i ≠ s, s + 1

cs(τ) = cs+1(σ) + 1

cs+1(τ) = cs(σ)otherwise ⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

ci(τ) = ci(σ) for i ≠ s, s + 1

cs(τ) = cs+1(σ)cs+1(τ) = cs(σ) − 1

Proof. (i) Follows definitions of the Lehmer code and σs.

(ii) Note first, that

τ = (σ(1), σ(2), . . . , σ(s − 1), σ(s + 1), σ(s), σ(s + 2), . . . , σ(n))Threrefore ci(σ) = ci(τ) for i ≠ s, s + 1 (for i < s we have: s ∈ C(σ)i if and only if

s + 1 ∈ C(τ)i).Suppose that σ(s) < σ(s + 1). In this case

C(τ)s = C(σ)s+1 ∪ {s + 1} and C(τ)s+1 = C(σ)s

6 P. Zawiślak

If σ(s) > σ(s + 1), then

C(τ)s = C(σ)s+1 and C(τ)s+1 = C(σ)s ∖ {s + 1}This finishes the proof.

Now we are ready to define the Lehmer factorial norm on Sn.

Definition 3.4. Let σ ∈ Sn be a permutation with the Lehmer code

lc(σ) = [c1(σ), c2(σ), . . . , cn(σ)] = [kn−1(σ), kn−2(σ), . . . , k0(σ)](here ki(σ) = cn−i(σ) for i = 0,1, . . . , n − 1). The Lehmer factorial norm (with base 2)

LF2 ∶ Sn → N is given by

LF2(σ) = n−1

∑i=0

[2i − 2i−ki(σ)]Remark 3.5. For a number m ∈ N let

m = kn−1 ⋅ (n − 1)! + . . . + k1 ⋅ 1! + k0 ⋅ 0!

be the (unique!) decomposition of m in such a way, that 0 ≤ ki ≤ i! for i = 0,1, . . . , n−

1 (in particular k0 = 0). Therefore m has the following factorial number system

representation

kn−1 ∶ . . . ∶ k1 ∶ 0!

Consider the function LF2 ∶ N→ N given by

LF2(m) = LF2 (kn−1 ⋅ (n − 1)! + . . . + k1 ⋅ 1! + k0 ⋅ 0!) = n−1

∑i=0

[2i− 2i−ki]

For σ ∈ N let nlex(σ) be the position of σ in the lexicographical order (numbering

starting from 0). Then

LF2(σ) = LF2(nlex(σ))The values LF2(σ) for σ ∈ S3 are presented in Table 1.

The next theorem yields information about the basic properties of the Lehmer

norm.


Theorem 3.6. The norm LF2 satisfies the following:

(i) LF2(en) = 0 is minimal and en (the identity) is the only permutation with this

property.

(ii) LF2(en) = 2n− (n + 1) is maximal and en (the reverse of the identity) is the

only permutation with this property.

(iii) LF2(σs) = 2n−1−s (recall that σs denotes the adjacent transposition) for s =1,2, . . . , n − 1, and therefore

LF2(σ1) > LF2(σ2) > . . . > LF2(σn−1)(iv) The inclusion ιn ∶ Sn → Sn+1 given by

ιn(σ) = (1, σ(1) + 1, σ(2) + 1, . . . , σ(n) + 1)preserves LF2.

(v) LF2(σ) = LF2(σ−1) for all σ ∈ Sn.

(vi) LF2(στ) ≤ LF2(σ) +LF2(τ) for all σ, τ ∈ Sn.

(vii) Let τ = σσs. Then

∣LF2(τ) −LF2(σ)∣ = 2−min{cs(σ),cs+1(σ)}LF2(σs)Proof. (i) Note that LF2(σ) ≥ 0 with the equality holds only if ki(σ) = 0 for i =0,1, . . . , n − 1. In such a case σ = en.

(ii) The proof is similar to the one of (i). Namely, LF2(σ) is maximal only if

ki(σ) = i for i = 0,1, . . . , n − 1 and this implies σ = en.

(iii) It is enough to see that ci(σs) = δis (see Lemma 3.3 (i)).

(iv) Follows the fact that for σ = (σ(1), σ(2), . . . , σ(n)) and ιn(σ) = (1,1+σ(1),1+σ(2), . . . ,1+σ(n)) we have c1(ιn(σ)) = 0 and ci(ιn(σ)) = ci−1(σ) for i = 2,3, . . . , n+1.

(v) First note, that

LF2(σ) = n−1

∑i=0

[2i− 2i−ki(σ)] = n−1

∑i=0

[2i− 2i−cn−i(σ)] = n

∑j=1

[2n−j− 2n−j−cj(σ)]

8 P. Zawiślak

Consequently, the equality LF2(σ) = LF2(σ−1) is equivalent to

n

∑j=1

2n−j−cj(σ) =n

∑j=1

2n−j−cj(σ−1)

Now according to Corollary 3.2

n − j − cj(σ−1) = n − j − [cσ−1(j)(σ) + σ−1(j) − j] = n − σ−1(j) − cσ−1(j)(σ)hence it is enough to notice that

n

∑j=1

2n−j−cj(σ) =n

∑j=1

2n−σ−1(j)−cσ−1(j)(σ)

is just change of order of summation. The last equality holds since for j taking all

values from [n] the same holds for σ−1(j).(vi) We have the following equalities:

LF2(σ) = n

∑j=1

[2n−j− 2n−j−cj(σ)]

LF2(τ) = n

∑j=1

[2n−j− 2n−j−cj(τ)]

and

LF2(στ) = n

∑j=1

[2n−j− 2n−j−cj(στ)]

Therefore the inequality

LF2(στ) ≤ LF2(σ) +LF2(τ)is equivalent to the following ones

n

∑j=1

[2n−j− 2n−j−cj(στ)] ≤ n

∑j=1

[2n−j− 2n−j−cj(σ)] + n

∑j=1

[2n−j− 2n−j−cj(τ)]

n

∑j=1

2n−j−cj(τ)+

n

∑j=1

2n−j−cj(σ) ≤n

∑j=1

2n−j+

n

∑j=1

2n−j−cj(στ)

n

∑j=1

1

2j+cj(τ)+

n

∑j=1

1

2j+cj(σ)≤

n

∑j=1

1

2j+

n

∑j=1

1

2j+cj(στ)


n

∑j=1

1

2j+cj(τ)+

n

∑j=1

1

2τ(j)+cτ(j)(σ)≤

n

∑j=1

1

2τ(j)+

n

∑j=1

1

2j+cj(στ)

The last inequality holds since for j taking all values from [n] the same holds for

τ(j).To finish the proof, it is enough to show that for every j = 1,2, . . . , n we have

(3.1)1

2j+cj(τ)+

1

2τ(j)+cτ(j)(σ)≤

1

2τ(j)+

1

2j+cj(στ)

By Lemma 3.1 (i) and (ii),

(3.2) cj(στ) ≤ cj(τ) + cτ(j)(σ) and τ(j) ≤ j + cj(τ).Since for non negative numbers a, b, c, d and e satisfying

e ≤ b + d and c ≤ a + b

it holds1

2a+b+

1

2c+d≤

1

2c+

1

2a+e

hence (3.1) is a consequence of (3.2) by substitution

a = j, b = cj(τ), c = τ(j), d = cτ(j)(σ) and e = cj(στ)(vii)

LF2(τ) −LF2(σ) = n−1

∑i=0

[2i− 2i−ki(τ)] − n−1

∑i=0

[2i− 2i−ki(σ)] = n−1

∑i=0

[2i−ki(σ)− 2i−ki(τ)] =

=n−1

∑i=0

[2i−cn−i(σ)− 2i−cn−i(τ)] = n

∑j=1

[2n−j−cj(σ)− 2n−j−cj(τ)]

Now according to Lemma 3.3 (ii) cj(τ) = cj(σ) for j ≠ s, s + 1, hence

LF2(τ) −LF2(σ) = [2n−s−cs(σ)− 2n−s−cs(τ)] + [2n−s−1−cs+1(σ)

− 2n−s−1−cs+1(τ)]If σ(s) < σ(s + 1), then by Lemma 3.3 (ii)

cs(τ) = cs+1(σ) + 1 and cs+1(τ) = cs(σ)

10 P. Zawiślak

Therefore

LF2(τ) −LF2(σ) = [2n−s−cs(σ)− 2n−s−cs+1(σ)−1] + [2n−s−1−cs+1(σ)

− 2n−s−1−cs(σ)] == 2n−s−1−cs(σ) = 2−cs(σ)LF2(σs)

Otherwise

cs(τ) = cs+1(σ) and cs+1(τ) = cs(σ) − 1

and therefore

LF2(τ) −LF2(σ) = [2n−s−cs(σ)− 2n−s−cs+1(σ)] + [2n−s−1−cs+1(σ)

− 2n−s−1−cs(σ)+1] == −2n−s−1−cs+1(σ) = −2−cs+1(σ)LF2(σs)

The last equalities in both cases are due to (iii).

To finish the proof, it is enough to notice that: if σ(s) < σ(s + 1), then cs(σ) ≤cs+1(s), otherwise cs+1(σ) < cs(σ).

4 The distribution of the Lehmer factorial norm

In this section we examine the properties of the probability distribution function of

the values of the Lehmer norm.

We start with the following theorem, the proof of which is due to K. Majcher.

Theorem 4.1. The direct limit of the system of groups (Sn, ιn) is given by

limÐ→ (Sn, ιn) ≅ S∞

Proof. Note first, that

S∞ ≅ limÐ→ (Sn, jn)where jn ∶ Sn → Sn+1 is given by

jn(σ(1), σ(2), . . . , σ(n)) = (σ(1), σ(2), . . . , σ(n), n + 1)To see this for σ ∈ S∞ let K(σ) be a minimal natural number such that σ(k) = k for

all k ≥K(σ). Define

F (σ) = ⎧⎪⎪⎨⎪⎪⎩(1) ∈ S1 if K(σ) = 1

(σ(1), . . . , σ(K(σ) − 1)) ∈ SK(σ)−1 if K(σ) > 2


(note, that it is impossible to have K(σ) = 2). F (σ) determines the unique element

G(σ) = (F (σ), jK(σ)−1(F (σ)), . . .)∼ ∈ limÐ→ (Sn, jn)It is easy to see that G is the isomorphism between S∞ and limÐ→ (Sn, jn).

To finish the proof it is enough to note that for tn being the conjugacy by en the

following diagrams commutes:

Sn

ιnÐÐÐ→ Sn+1

tn

×××Ö×××Ötn+1

Sn ÐÐÐ→jn

Sn+1

Note, that due to Theorems 4.1 and 3.6 (iv), LF2 can be seen as a norm on S∞.

We continue with the following observation concerning the properties of permu-

tations from the image ιn−1[Sn−1]. According to the Theorem 3.6 (ii) and (iv) we

have the following:

Remark 4.2. Let σ ∈ Sn be a permutation. If c1(σ) = 0, then

LF2(σ) ≤ 2n−1− n

On the other hand, if c1(σ) > 0, then

LF2(σ) ≥ 2n−2

The following definition will be crucial to dermine the distribution of LF2 on S∞.

Definition 4.3. For a natural number m > 0 and for k = 0,1, . . . let

Sk(m) = ⋃t≥1

{((m1, l1), (m2, l2), . . . , (mt, lt)) ∈ N2t∶ k =m1 >m2 > . . . >mt ≥ 0;

mj ≥ lj ≥ 0 for j = 1,2, . . . , t; m =t

∑j=1

⎡⎢⎢⎢⎢⎣lj

∑p=0

2mj−p

⎤⎥⎥⎥⎥⎦}

and let sk(m) = ∣Sk(m)∣.

12 P. Zawiślak

Lemma 4.4. Let m = ∑sj=1 2mj for some natural numbers m1 > m2 > . . . > ms ≥ 0.

Then sk(m) = 0 for all k ≠m1,m1 − 1.

Proof. Suppose first that k >m1. Therefore

m =s

∑j=1

2mj ≤m1

∑i=0

2i = 2m1+1− 1 < 2k

hence m cannot be decomposed as a sum given by any element

((m1, l1), (m2, l2), . . . , (mt, lt)) ∈ Sk(m)On the other hand, if k <m1 − 1, then (for m1 = k)

t

∑j=1

⎡⎢⎢⎢⎢⎣lj

∑p=0

2mj−p

⎤⎥⎥⎥⎥⎦≤

k

∑i=0

[ i

∑r=0

2r] = k

∑i=0

[2i+1− 1] = 2k+2

− (k + 2) < 2k+2 ≤ 2m1 ≤m

and similarily, m cannot be decomposed as a sum given any element

((m1, l1), (m2, l2), . . . , (mt, lt)) ∈ Sk(m)

According to Lemma 4.4

s(m) = ∞∑k=0

sk(m)is a well defined natural number. Moreover, the following holds:

Theorem 4.5. Put s(0) = 1. Then for every natural number m we have

s(m) = ∣{σ ∈ S∞ ∶ LF2(σ) =m}∣Proof. The statement is obvious for m = 0.

Consider σ ∈ S∞ different from the identity. Let n be the minimal natural number

such that σ can be regarded as an element of Sn. Since σ is not the identity, it follows

that n > 1. Finally, let m = LF2(σ). Therefore

m = LF2(σ) = n−1

∑i=0

[2i− 2i−ki(σ)] = n−1

∑i=0

ki(σ)≠0

⎡⎢⎢⎢⎢⎣ki(σ)−1

∑p=0

2(i−1)−p

⎤⎥⎥⎥⎥⎦=

n−1

∑i=1

ki(σ)≠0

⎡⎢⎢⎢⎢⎣ki(σ)−1

∑p=0

2(i−1)−p

⎤⎥⎥⎥⎥⎦


Since n is minimal, it follows that kn−1(σ) = c1(σ) > 0.

Let i1 > i2 > . . . > it be all elements of [n−1] such that kij(σ) ≠ 0 for j = 1,2, . . . , t

(of course t ≥ 1) and let mj = ij − 1. Thus we have

n − 2 =m1 >m2 > . . . >mt ≥ 0

Put lj = kij(σ) − 1 and note, that lj ≤mj .

Since

m =n−1

∑i=1

ki(σ)≠0

⎡⎢⎢⎢⎢⎣ki(σ)−1

∑p=0

2(i−1)−p

⎤⎥⎥⎥⎥⎦=

t

∑j=1

⎡⎢⎢⎢⎢⎣lj

∑p=0

2mj−p

⎤⎥⎥⎥⎥⎦this decompostition of LF2(σ) =m determines the element

((m1, l1), (m2, l2), . . . , (mt, lt)) ∈ Sn−2(m)On the other hand, for an element

((m1, l1), (m2, l2), . . . , (mt, lt)) ∈ Sn(m)let σ ∈ Sn+2 be given by its Lehmer code in the following way:

lc(σ) = [c1(σ), c2(σ), . . . , cn+2(σ)] = [kn+1(σ), kn(σ), . . . , k0(σ)]where

• kmj+1(σ) = lj + 1 for j = 1,2, . . . , t,

• ki(σ) = 0 for i ∈ {0,1, . . . ,m + 1} ∖ {m1 + 1,m2 + 1, . . . ,mt + 1}For σ defined in such a way it holds

LF2(σ) = n+1

∑i=0

[2i− 2i−ki(σ)] = n+1

∑i=0

ki(σ)≠0

⎡⎢⎢⎢⎢⎣ki(σ)−1

∑p=0

2(i−1)−p

⎤⎥⎥⎥⎥⎦=

=t

∑j=1

⎡⎢⎢⎢⎢⎣kmj+1(σ)−1

∑p=0

2[(mj+1)−1]−p

⎤⎥⎥⎥⎥⎦=

t

∑j=1

⎡⎢⎢⎢⎢⎣lj

∑p=0

2mj−p

⎤⎥⎥⎥⎥⎦=m

Moreover, the assigments

σ → ((m1, l1), (m2, l2), . . . , (mt, lt)) and ((m1, l1), (m2, l2), . . . , (mt, lt)) → σ

14 P. Zawiślak

are mutually inverse.

This finishes the proof.

In the next theorem we present the recursive properties of numbers sk(m). Note,

that Theorems 4.5 and 4.6 together with Lemma 4.4 fully describe the distribution of

LF2 on S∞. The values of sk(m) for m = 1,2, . . . ,8, together with the corresponding

permutations can be found in Table 2. The graphs of functions s(m) and d(m) =∑m

l=1 s(l) for m = 1,2, . . . ,256 are presented in Figure 1.

Theorem 4.6. The following holds:

(i) s(0) = 1;

(ii) s0(1) = 1 and sk(1) = 0 for k > 0;

(iii) s0(2) = 0, s1(2) = 1 and sk(2) = 0 for k > 1;

(iv) sm (2m) = 1,

sm−1 (2m) = m−1

∑j=0

[m−2

∑k=0

sk (2m− (2m−1

+ . . . + 2j))]and sk (2m) = 0 for k ≠m,m − 1;

(v) For l = 0,1, . . . ,m − 1

sm (2m+ . . . + 2l) = 1 +

m−1

∑j=l

[m−1

∑k=0

sk (2j+ . . . + 2l)] ,

sm−1 (2m+ . . . + 2l) = m−1

∑j=0

[m−2

∑k=0

sk ((2m+ . . . + 2l) − (2m−1

+ . . . + 2j))]and sk (2m

+ . . . + 2l) = 0 for k ≠m,m − 1;

(vi) For l = 2,3, . . . ,m − 1 and for a0, a1, . . . , al−2 ∈ {0,1} not all being equal to 0

sm (2m+ . . . + 2l

+ al−22l−2+ . . . + a02

0) =m−1

∑k=0

[sk (al−22l−2+ . . . + a020)] + m−1

∑j=l

[m−1

∑k=0

sk (2j+ . . . + 2l

+ al−22l−2+ . . . + a020)] ,


sm−1 (2m+ . . . + 2l

+ al−22l−2+ . . . + a020) =

=m−1

∑j=0

[m−2

∑k=0

sk ((2m+ . . . + 2l

+ al−22l−2+ . . . + a020) − (2m−1

+ . . . + 2j))]and sk (2m

+ . . . + 2l+ al−22l−2

+ . . . + a020) = 0 for k ≠m,m − 1;

(vii) For m > 1 and for a0, a1, . . . , am−2 ∈ {0,1} not all being equal to 0

sm (2m+ am−22m−2

+ . . . + a020) = m−1

∑k=0

[sk (am−22m−2+ . . . + a020)] ,

sm−1 (2m+ am−22m−2

+ . . . + a020) ==

m−1

∑j=0

[m−2

∑k=0

sk ((2m+ am−22m−2

+ . . . + a020) − (2m−1

+ . . . + 2j))]and sk (2m

+ am−22m−2+ . . . + a020) = 0 for k ≠m,m − 1.

Proof. (i) follows the definition of s.

(ii) and (iii) follows the equations S0(1) = {(0,0)} and S1(2) = {(1,0)} respec-

tively, as well as Lemma 4.4.

(iv) There are three cases to consider when determining Sk(2m), namely k = m,

k =m − 1 and k ≠m,m − 1.

(a) Sm(2m) = {(m,0)}, hence sm(2m) = 1.

(b) Consider first an element

((m1, l1), (m2, l2), . . . , (mt, lt)) ∈ Sm−1(2m)Therefore m1 =m − 1. Putting j =m − 1 − l1 we get m − 1 ≥ j ≥ 0. And since

2m >m−1

∑i=j

2i =l1

∑p=0

2(m−1)−p =l1

∑p=0

2m1−p

one must have t > 1. Therefore (note that m2 ≤m − 2)

((m2, l2), . . . , (mt, lt)) ∈ m−2

⊍k=0

Sk (2m− ( l1

∑p=0

2m1−p)) = m−2

⊍k=0

Sk (2m− (2m−1

+ . . . + 2j))

16 P. Zawiślak

Conversely, for every j = 0,1, . . . ,m − 1 an element

((m2, l2), . . . , (mt, lt)) ∈ m−2

⊍k=0

Sk (2m− (2m−1

+ . . . + 2j))determines

((m − 1,m − 1 − j), (m2, l2), . . . , (mt, lt)) ∈ Sm−1(2m)(c) The equality sk (2m) = 0 for k ≠m,m − 1 follows Lemma 4.4.

(v) There are three cases to consider when determining Sk(2m+ . . . + 2l), namely

k =m, k =m − 1 and k ≠m,m − 1.

(a) Consider first an element

((m1, l1), (m2, l2), . . . , (mt, lt)) ∈ Sm(2m+ . . . + 2l)

If t = 1, then (m1, l1) = (m,m − l).Otherwise l1 <m − l. To see this note that for l1 ≥m − l we have

[2m+ . . . + 2m−l1] + [2m2

+ . . . + 2m2−l2] ≥ [2m+ . . . + 2l] + [2m2] > 2m

+ . . . + 2l

hence

((m1, l1), (m2, l2), . . . , (mt, lt)) ∉ Sm(2m+ . . . + 2l)

Let j =m − 1 − l1 (in particular m − 1 ≥ j > l − 1). Now

2m+ . . . + 2l >

m

∑i=j+1

2i =l1

∑p=0

2m−p =l1

∑p=0

2m1−p

and therefore (since m2 ≤m − 1)

((m2, l2), . . . , (mt, lt)) ∈ m−1

⊍k=0

Sk ((2m+ . . . + 2l) − l1

∑p=0

2m1−p) = m−1

⊍k=0

Sk (2j+ . . . + 2l)

Conversely, for every j = l, . . . ,m − 1 an element

((m2, l2), . . . , (mt, lt)) ∈ m−1

⊍k=0

Sk (2j+ . . . + 2l)

defines

((m,m − 1 − j), (m2, l2), . . . , (mt, lt)) ∈ Sm (2m+ . . . + 2l)

Together with (m,m − l) these are all elements of Sm (2m+ . . . + 2l).



((m1, l1), (m2, l2), . . . , (mt, lt)) ∈ Sm−1(2m+ . . . + 2l)

Therefore m1 =m − 1. Putting j =m − 1 − l1 we get m − 1 ≥ j ≥ 0. Since

2m+ . . . + 2l >

m−1

∑i=j

2i =l1

∑p=0

2(m−1)−p =l1

∑p=0

2m1−p

it follows that t > 1. Therefore (note that m2 ≤m − 2)

((m2, l2), . . . , (mt, lt)) ∈ m−2

⊍k=0

Sk ((2m+ . . . + 2l) − ( l1

∑p=0

2m1−p)) ==

m−2

⊍k=0

Sk ((2m+ . . . + 2l) − (2m−1

+ . . . + 2j))Conversely, for every j = 0,1, . . . ,m − 1, an element

((m2, l2), . . . , (mt, lt)) ∈ m−2

⊍k=0

Sk ((2m+ . . . + 2l) − (2m−1


((m − 1,m − 1 − j), (m2, l2), . . . , (mt, lt)) ∈ Sm−1 (2m+ . . . + 2l)

(c) The equality sk (2m+ . . . + 2l) = 0 for k ≠m,m − 1 follows Lemma 4.4.

(vi) There are three cases to consider when determining Sk(2m+ . . . + 2l

+ al−22l−2+

. . . + a020), namely k =m, k =m − 1 and k ≠m,m − 1.


((m1, l1), (m2, l2), . . . , (mt, lt)) ∈ Sm(2m+ . . . + 2l

+ al−22l−2+ . . . + a020)

For l1 >m − l we have

2m+ . . . + 2m−l1 ≥ 2m

+ . . . + 2l+ 2l−1 > 2m

+ . . . + 2l+ al−22

l−2+ . . . + a02

2

and therefore

((m1, l1), (m2, l2), . . . , (mt, lt)) ∉ Sm(2m+ . . . + 2l

+ al−22l−2+ . . . + a020)

18 P. Zawiślak

hence l1 ≤m − l. And since

l1

∑p=0

2m1−p ≤m−l

∑p=0

2m1−p =m−l

∑p=0

2m−p < 2m+ . . . + 2l

+ al−22l−2+ . . . + a020

it follows that t > 1.

If l1 =m − l, thenl1

∑p=0

2m1−p = 2m+ . . . + 2l


((m2, l2), . . . , (mt, lt)) ∈ m−1

⊍k=0

Sk (al−22l−2+ . . . + a020)

If l1 <m − l, putting j =m − 1 − l1 we get l ≤ j ≤m − 1. In this case

l1

∑p=0

2m1−p =m−1−j

∑p=0

2m−p = 2m+ . . . + 2j+1

thus (since m2 ≤m − 1)

((m2, l2), . . . , (mt, lt)) ∈ m−1

⊍k=0

Sk (2j+ . . . + 2l

+ al−22l−2+ . . . + a020)

Conversely, an element

((m2, l2), . . . , (mt, lt)) ∈ m−1

⊍k=0

Sk (al−22l−2+ . . . + a020)

determines

((m,m − l), (m2, l2), . . . , (mt, lt)) ∈ Sm(2m+ . . . + 2l

+ al−22l−2+ . . . + a020)

and, for j = l, . . . ,m − 1, an element

((m2, l2), . . . , (mt, lt)) ∈ m−1

⊍k=0

Sk (2j+ . . . + 2l

+ al−22l−2+ . . . + a020)

determines

((m,m − 1 − j), (m2, l2), . . . , (mt, lt)) ∈ Sm(2m+ . . . + 2l

+ al−22l−2+ . . . + a020)



((m1, l1), (m2, l2), . . . , (mt, lt)) ∈ Sm−1(2m+ . . . + 2l

+ al−22l−2+ . . . + a02

0)Since m1 =m − 1, we get l1 ≤m − 1. Therefore

l1

∑p=0

2m1−p ≤m−1

∑p=0

2m−1−p < 2m < 2m+ . . . + 2l

+ al−22l−2+ . . . + a020

and hence t > 1. Putting j =m1 − l1 we get 0 ≤ j ≤m − 1. Now

l1

∑p=0

2m1−p =m−1−j

∑p=0

2m−1−p = 2m−1+ . . . + 2j


((m2, l2), . . . , (mt, lt)) ∈ m−2

⊍k=1

Sk ((2m+ . . . + 2l

+ al−22l−2+ . . . + a020) − (2m−1

+ . . . + 2j))Conversely, for every j = 0,1, . . . , k − 1 an element

((m2, l2), . . . , (mt, lt)) ∈ m−2

⊍k=0

Sk ((2m+ . . . + 2l

+ al−22l−2+ . . . + a020) − (2m−1


((m− 1,m− 1− j), (m2, l2), . . . , (mt, lt)) ∈ Sm−1(2m+ . . .+ 2l

+al−22l−2+ . . .+a02

0)(c) The equality sk (2m

+ . . . + 2l+ al−22l−2

+ . . . + a020) = 0 for k ≠ m,m − 1 follows

Lemma 4.4.

(vii) There are three cases to consider when determining Sk(2m+am−22m−2

+. . .+a020),namely k =m, k =m − 1 and k ≠m,m − 1.


((m1, l1), (m2, l2), . . . , (mt, lt)) ∈ Sm(2m+ am−22m−2

+ . . . + a020)For l1 > 0 we have

l1

∑p=0

2m1−p ≥ 2m+ 2m−1 > 2m

+ am−22m−2+ . . . + a020

20 P. Zawiślak

and therefore

((m1, l1), (m2, l2), . . . , (mt, lt)) ∉ Sm(2m+ am−22m−2

+ . . . + a020)hence l1 = 0 and t > 1. Therefore (since m2 <m)

((m2, l2), . . . , (mt, lt)) ∈ m−1

⊍k=0

Sk(al−22l−2+ . . . + a020)

Conversely, an element

((m2, l2), . . . , (mt, lt)) ∈ m−1

⊍k=0

Sk(am−22m−2+ . . . + a02

0)determines

((m,0), (m2, l2), . . . , (mt, lt)) ∈ Sm(2m+ am−22m−2

+ . . . + a020)


((m1, l1), (m2, l2), . . . , (mt, lt)) ∈ Sm−1(2m+ am−22m−2

+ . . . + a020)Since m1 =m − 1, we get l1 ≤m − 1. Therefore

l1

∑p=0

2m1−p ≤m−1

∑p=0

2m−1−p < 2m < 2m+ am−22m−2

+ . . . + a020

and hence t > 1. Putting j =m1 − l1 we get 0 ≤ j ≤m − 1. Now

l1

∑p=0

2m1−p =m−1−j

∑p=0

2m−1−p = 2m−1+ . . . + 2j


((m2, l2), . . . , (mt, lt)) ∈ m−2

⊍k=1

Sk ((2m+ am−22m−2

+ . . . + a020) − (2m−1+ . . . + 2j))

Conversely, for every j = 0,1, . . . , k − 1, an element

((m2, l2), . . . , (mt, lt)) ∈ m−2

⊍k=0

Sk ((2m+ am−22m−2

+ . . . + a020) − (2m−1+ . . . + 2j))

determines

((m − 1,m − 1 − j), (m2, l2), . . . , (mt, lt)) ∈ Sm−1(2m+ am−22m−2

+ . . . + a020)


(c) The equality sk (2m+ am−22m−2

+ . . . + a020) = 0 for k ≠ m,m − 1 follows Lemma

4.4.

This finishes the proof.

Acknowledgements

The author is grateful to K. Majcher for the proof of Theorem 4.1, to P. Józiak for

the improvement of the proof of Theorem 3.6 (vi) as well as for carefully reading of

this paper, and to J. Gismatullin for inspiring converations.

Last, but not least, the author wants to thank his whife for her strong and loving

support as well as for her inspiring ”try to think nonstandard”.

All calculations were performed with R 4.0.3 ([RPackage]).

During the work on this paper the author was partially supported by the SGH

fund KAE/S21 and by the NCN fund UMO-2018/31/B/HS4/01005.

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24 P. Zawiślak

Table 1: The Lehmer factorial norm on S3

nlex(σ) σ lc(σ) LF2(σ) nlex(σ) in the factorial number system representation0 (1,2,3) [0,0,0] 0 0 = 0 ⋅ 2! + 0 ⋅ 1! + 0 ⋅ 0!1 (1,3,2) [0,1,0] 1 1 = 0 ⋅ 2! + 1 ⋅ 1! + 0 ⋅ 0!2 (2,1,3) [1,0,0] 2 2 = 1 ⋅ 2! + 0 ⋅ 1! + 0 ⋅ 0!3 (2,3,1) [1,1,0] 3 3 = 1 ⋅ 2! + 1 ⋅ 1! + 0 ⋅ 0!4 (3,1,2) [2,0,0] 3 4 = 2 ⋅ 2! + 0 ⋅ 1! + 0 ⋅ 0!5 (3,2,1) [2,1,0] 4 5 = 2 ⋅ 2! + 1 ⋅ 1! + 0 ⋅ 0!

Table 2: The elements of Sk(m)m decomposition of m lc(σ) ((m1, l1), . . . , (mt, lt)) element of1 1 = [20] [1,0] (0,0) S0(1)2 2 = [21] [1,0,0] (1,0) S1(2)3 3 = [21] + [20] [1,1,0] ((1,0), (0,0)) S1(3)3 3 = [21

+ 20] [2,0,0] (1,1) S1(3)4 4 = [21

+ 20] + [20] [2,1,0] ((1,1), (0,0)) S1(4)4 4 = [22] [1,0,0,0] (2,0) S2(4)5 5 = [22] + [20] [1,0,1,0] ((2,0), (0,0)) S2(5)6 6 = [22] + [21] [1,1,0,0] ((2,0), (1,0)) S2(6)6 6 = [22

+ 21] [2,0,0,0] (2,1) S2(6)7 7 = [22] + [21] + [20] [1,1,1,0] ((2,0), (1,0), (0,0)) S2(7)7 7 = [22] + [21

+ 20] [1,2,0,0] ((2,0), (1,1)) S2(7)7 7 = [22

+ 21] + [20] [2,0,1,0] ((2,1), (0,0)) S2(7)7 7 = [22

+ 21+ 20] [3,0,0,0] (2,2) S2(7)

8 8 = [22] + [21+ 20] + [20] [1,2,1,0] ((2,0), (1,1), (0,0)) S2(8)

8 8 = [22+ 21] + [21] [2,1,0,0] ((2,1), (1,0)) S2(8)

8 8 = [22+ 21+ 20] + [20] [3,0,1,0] ((2,2), (0,0)) S2(8)

8 8 = [23] [1,0,0,0,0] (3,0) S3(8)


Figure 1: The graphs of functions s(m) and d(m)

0 50 100 150 200 250 300

0

200

400

600

m

s(m)

Function s(m)

0 50 100 150 200 250 300

0

10,000

20,000

30,000

40,000

50,000

m

d(m)=∑

m l=1s(l)

Function d(m)

The Lehmer factorial norm on Sn

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