Discrete Applied Mathematics

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Discrete Applied Mathematics

journal homepage:www.elsevier.com/locate/dam

A permutation code preserving a double Eulerian bistatistic

Jean-Luc Baril, Vincent Vajnovszki

^∗

LE2I, Université de Bourgogne Franche-Comté, BP 47870, 21078 Dijon Cedex, France

a r t i c l e i n f o

Article history:

Received 19 October 2016

Received in revised form 10 February 2017 Accepted 16 February 2017

Available online 22 March 2017

Keywords:

Set-valued/Eulerian statistic on permutation

Permutation/Lehmer code Subexcedant sequence Constructive bijection

a b s t r a c t

In 1977 Foata proved bijectively, among other things, that the joint distribution of ascent and distinct nonzero value numbers on the set of subexcedant sequences is the same as that of descent and inverse descent numbers on the set of permutations, and the generating function of the corresponding bistatistics is the double Eulerian polynomial. In 2013 Foata’s result was rediscovered by Visontai as a conjecture, and then reproved by Aas in 2014.

In this paper, we define a permutation code (that is, a bijection between permutations and subexcedant sequences) and show the more general result that two 5-tuples of set- valued statistics on the set of permutations and on the set of subexcedant sequences, respectively, are equidistributed. In particular, these results give another bijective proof of Foata’s result.

©2017 Published by Elsevier B.V.

1. Introduction

In enumerative combinatorics, it is a classical result that the descent numberdesand the inverse descent numberides (defined asides

π =

des

π

⁻¹) on permutations are Eulerian statistics, and their distributions on the setS_nof length-n permutations are given by thenth Eulerian polynomialA_n, that is

A_n

(

^u

) =



π∈Sn

u^desπ+1

=



π∈Sn

u^idesπ+1

,

and the joint distribution ofdesandidesis given by thenth double Eulerian polynomial, A_n

(

^u

, v) =



π∈Sn

u^desπ+1

v

^idesπ+1

,

see for instance [2,8].

An alternative way to represent a permutation is its Lehmer code [5], which is a subexcedant sequence. The ascent numberascon the setS_nof subexcedant sequences is still an Eulerian statistic (see for example [9]), and in [7] the statistic that counts the number of distinct nonzero symbols in a subexcedant sequence (that following [1] we denote byrow) is proved to be still Eulerian; this result is credited to Dumont by the authors of [7]. In terms of generating functions, we have

A_n

(

^u

) =



s∈Sn

u^ascs+1

=



s∈Sn

u^rows+1

.

∗Corresponding author.

E-mail addresses:[email protected](J.-L. Baril),[email protected](V. Vajnovszki).

http://dx.doi.org/10.1016/j.dam.2017.02.014 0166-218X/©2017 Published by Elsevier B.V.

In 1977 Foata [4] proved, among other things, that the joint distribution ofdesandideson the set of permutations is the same as that ofascandrowon the set of subexcedant sequences, that is

A_n

(

^u

, v) =



π∈Sn

u^desπ+1

v

^idesπ+1

=



s∈Sn

u^ascs+1

v

^rows+1

.

In the present paper, we define a bijection between permutations and subexcedant sequences (i.e., a permutation code) and show that the tuple of set-valued statistics

(

^Des

,

^Ides

,

^Lrmax

,

^Lrmin

,

^Rlmax

)

on the set of permutations, and

(

^Asc

,

^Row

,

^Posz

,

^Max

,

^Rlmax

)

on the set of subexcedant sequences

have the same distribution (each of the occurring statistics is defined below). In particular, our bijection provides set-valued partners forAsc(answering to a question in [1]) and gives a alternative proof of Foata’s result.

2. Notation and definitions

A length-n word

w

over the alphabetAis a sequence

w

1

w

2

. . . w

nof symbols inA, and we will consider only finite alphabetsA

⊂

_N.

Statistics

Astatisticon a setXof words is simply a function fromXtoN; aset-valued statistic is a function fromX to 2^N; and a multistatisticis a tuple of statistics.

Let

w = w

1

w

2

. . . w

nbe a length-nword. Adescentin

w

is a positioniin

w

^{, 1}

≤

i

<

^{n, with}

w

i

> w

i+1, and thedescent setof

w

^is

Des

w = {

i

:

1

≤

i

<

^nwith

w

i

> w

i+1

} .

Aleft-to-right maximumin

w

is a positioniin

w

^{, 1}

≤

i

≤

n, with

w

j

< w

ifor allj

<

^{i, and the}set of left-to-right maximais Lrmax

w = {

i

:

1

≤

i

≤

nwith

w

j

< w

ifor allj

<

ⁱ

} .

Clearly, 1

∈

Lrmax

w

^{, andDesandLrmax}are classical examples of set-valued statistics on words. We define similarly the setsAsc

w

^{ofascents,Lrmin}

w

^ofleft-to-right minima,Rlmax

w

^ofright-to-left maximaandRlmin

w

^ofright-to-left minimain

w

^.

To each set-valued statisticStcorresponds an (integer-valued) statisticstdefined asst

w =

cardSt

w

, for exampledes

w

andlrmax

w

count, respectively, the number of descents and the number of left-to-right maxima in

w

^.

LetXandX^′be two sets of words, andstandst^′be two statistics defined onXandX^′, respectively. We say thatstonX has thesame distributionasst^′onX^′(or equivalently,standst^′areequidistributed) if, for any integeru,

card

{ w ∈

X

:

st

w =

u

} =

card

{ w ∈

X^′

:

st^′

w =

u

} ,

and the multistatistic

(

^st1

,

^st2

, . . . ,

^stp

)

^{defined onX}has the same distribution as the multistatistic

(

^st′1

,

^st′2

, . . . ,

^st′p

)

^defined onX^′(or the two multistatistics are equidistributed) if, for any integerp-tupleu

= (

^u1

,

^u2

, . . . ,

^up

)

^,

card

{ w ∈

X

: (

^st1

,

^st2

, . . . ,

^stp

) w =

u

} =

card

{ w ∈

X^′

: (

^st′1

,

^st′2

, . . . ,

^st′p

) w =

u

} .

The notion of equidistribution of (multi)statistics can naturally be extended to set-valued (multi)statistics.

Permutations, subexcedant sequences and codes

This paper deals with two particular classes of words: permutations and subexcedant sequences. A permutation is a length- nword over

{

₁

,

²

, . . . ,

ⁿ

}

with distinct symbols. Alternatively, a permutation is an element of the symmetric group on

{

1

,

²

, . . . ,

ⁿ

}

written in one line notation, andS_ndenotes the set of length-npermutations. If two permutations

π = π

1

π

2

. . . π

nand

σ = σ

1

σ

2

. . . σ

nare such that

σ

π1

σ

π2

· · · σ

πn

=

12

· · ·

n(i.e., the identity inS_n), then

σ

^{is theinverseof}

π

, which is denoted by

π

^−1.

A length-n subexcedant sequence¹is a words

=

s₁s₂

. . .

^snover

{

0

,

¹

, . . . ,

ⁿ

−

1

}

with 0

≤

s_i

≤

i

−

1 for 1

≤

i

≤

n, and S_ndenotes the set of length-nsubexcedant sequences; and we haveS_n

= {

0

} × {

0

,

¹

} × · · · × {

0

,

¹

, . . . ,

ⁿ

−

1

}

.

Some statistics are consistently defined only on particular classes of words,e.g.permutations or subexcedant sequences.

For a permutation

π ∈

S_n, aninverse descent(idesfor short) in

π

is a positionifor which

π

i

+

1 appears to the left of

π

i

in

π

. Equivalently,iis an ides in

π

^if

π

iis a descent in

π

^{−1. Theides set}is defined as Ides

π = {

i

:

1

<

ⁱ

≤

nwith

π

i

+

1 appears in

π

to the left of

π

i

} ,

andides

π =

des

π

⁻¹, but in generalIdes

π

is not equal toDes

π

^−1.

Lets

=

s₁s₂

. . .

^snbe a subexcedant sequence inS_n. ThePoszstatistic gives the positions of 0s ins, Poszs

= {

i

:

1

≤

i

≤

n

,

^si

=

0

} ,

1 Known in the literature also asinversion sequence,inversion tableorsubexceedant function.

and obviously 1

∈

Poszs. TheMaxstatistic is defined as Maxs

= {

i

:

1

≤

i

≤

n

,

^si

=

i

−

1

} ,

and as above, 1

∈

Maxs.

Alast-value positioninsis a positioniinssuch thats_i

̸=

0 ands_idoes not occur in the suffixs_i+1s_i+2

. . .

^snofs. The last-value position set, denoted byRow, is defined as

Rows

= {

i

:

s_i

̸=

0 ands_idoes not occur in the suffixs_i+1s_i+2

. . .

^sn

} .

Clearly, 1

̸∈

Rows, androws

=

cardRowscounts the number of distinct nonzero symbols ins.

Example 1. If

π =

6 2 5 8 7 3 1 4

∈

S₈ands

=

0 1 1 0 2 3 6 3

∈

S₈, then Des

π =

Ascs

= {

1

,

⁴

,

⁵

,

⁶

}

,

Ides

π =

Rows

= {

3

,

⁵

,

⁷

,

⁸

}

, Lrmax

π =

Poszs

= {

1

,

⁴

}

, Lrmin

π =

Maxs

= {

1

,

²

,

⁷

}

, Rlmax

π =

Rlmins

= {

4

,

⁵

,

⁸

}

.

Aninversionin a permutation

π = π

1

π

2

. . . π

n

∈

S_nis a pair

(

ⁱ

,

^j

)

^withi

<

^jand

π

i

> π

j. The setS_nis in bijection with S_n, and any such bijection is calledpermutation code. TheLehmer code Ldefined in [5] is a classical example of permutation code; it maps each permutation

π = π

1

π

2

. . . π

nto a subexcedant sequences₁s₂

. . .

^snwhere, for allj, 1

≤

j

≤

n,s_jis the number of inversions

(

ⁱ

,

^j

)

ⁱⁿ

π

(or equivalently, the number of entries in

π

larger than

π

jand on its left). For example L

(

6 2 5 8 7 3 1 4

) =

0 1 1 0 1 4 6 4. See also [11] for a family of permutation codes in the context of Mahonian statistics on permutations.

In [3] is showed thatdmcstatistic which counts the number of distinct nonzero symbols in the Lehmer code of a permu- tation

π

(the statistic

π →

rowL

(π)

with the above notations) is Eulerian, and so has the same distribution asdes,ascor idesonS_n. See also [10] where Dumont’s statisticdmcis extended to words.

Although the following properties are folklore, they are easy to check.

Property 1. If

π ∈

S_nand L

(π) ∈

S_nis its Lehmer code, thenDes

π =

AscL

(π)

^,Lrmax

π =

PoszL

(π)

^,Lrmin

π =

MaxL

(π)

^, andRlmax

π =

RlminL

(π)

^.

3. The permutation codeb

We define a mappingb:S_n

→

S_nandTheorem 1shows thatbis a bijection, that is, a permutation code, and it is the main tool in proving that

(

^Des

,

^Ides

,

^Lrmax

,

^Lrmin

,

^Rlmax

)

^onSnhas the same distribution as

(

^Asc

,

^Row

,

^Posz

,

^Max

,

^Rlmax

)

^on S_n(seeTheorem 2).

A positioniin

π = π

1

π

2

. . . π

n

∈

S_n, 1

≤

i

≤

n, can satisfy the following properties:

P1:

π

i

+

1 occurs in

π

at the right of

π

i, P2:

π

i

−

1 occurs in

π

0 at the right of

π

i,

where

π

0 is the permutation of

{

0

,

¹

, . . . ,

ⁿ

}

obtained by adding a 0 at the end of

π

. And, to each positioniin

π

, we associate an integer

λ

i

(π) ∈ {

0

,

¹

,

²

,

³

}

according toisatisfies both, one, or none of these properties:

λ

i

(π) =







0

,

^ifisatisfies both P1 and P2

,

1

,

^ifisatisfies P2 but not P1

,

2

,

^ifisatisfies P1 but not P2

,

3

,

^ifisatisfies neither P1 nor P2

,

and we denote it simply by

λ

iwhen there is no ambiguity.

Alternatively, using the Iverson bracket notation (

[

P

] =

1 iff the statementP is true), we have the more concise expression:

λ

i

= [ π

i

=

nor

π

i

+

1 occurs at the left of

π

i

] +

2

· [ π

i

−

1 occurs at the left of

π

i

]

.

For example, for any

π ∈

S_nwe have

λ

1

=

0 except

λ

1

=

1 if

π

1

=

n; and whenn

>

^{1, then}

λ

n

=

3 except

λ

n

=

2 if

π

n

=

1. Each

λ

iis uniquely determined by

π

, for instance if

π =

6 2 5 8 7 3 1 4, then

λ

1

, λ

2

, . . . , λ

8

=

0

,

⁰

,

¹

,

¹

,

³

,

²

,

¹

,

^3, seeFig. 2.

Aninterval I

= [

a

,

^b

]

,a

≤

b, is the set of integers

{

x

:

a

≤

x

≤

b

}

; and alabeled intervalis a pair

(

^I

, ℓ)

^{whereIis an} interval and

ℓ

is an integer. In order to give the construction of the mappingb, we define below the slices of a permutation, and some of their properties are given inRemark 1.

Definition 1. For a permutation

π = π

1

π

2

. . . π

n

∈

S_nand ani, 0

≤

i

<

^{n, theithsliceof}

π

is the sequence of labeled intervalsU_i

(π) = (

^I1

, ℓ

1

), (

^I2

, ℓ

2

), . . . , (

^Ik

, ℓ

k

)

, defined by the following process (seeFig. 1).

Fig. 1. The four cases inDefinition 1.

•

U₀

(π) = ( [

0

,

ⁿ

] ,

⁰

)

^.

•

Fori

≥

1, letU_i−1

(π) = (

^I1

, ℓ

1

), (

^I2

, ℓ

2

), . . . , (

^Ik

, ℓ

k

)

^{be the}

(

ⁱ

−

1

)

th slice of

π

^and

v

^{, 1}

≤ v ≤

k, be the integer such that

π

i

∈

I_v. Theith sliceU_i

(π)

^of

π

is defined according to

λ

i:

– If

λ

i

=

0 (or equivalently, minI_v

< π

i

<

^maxIv), then

U_i

(π) = (

^I1

, ℓ

1

), . . . , (

^Iv−1

, ℓ

v−1

), (

^H

, ℓ

v

), (

^J

, ℓ

v+1

), (

^Iv+1

, ℓ

v+2

), . . . , (

^Ik−1

, ℓ

k

), (

^Ik

, ℓ

k

+

1

),

whereH

= [ π

i

+

1

,

^maxIv

]

andJ

= [

minI_v

, π

i

−

1

]

;

– If

λ

i

=

1 (or equivalently, minI_v

<

^maxIv

= π

i), then

U_i

(π) = (

^I1

, ℓ

1

), . . . , (

^Iv−1

, ℓ

v−1

), (

^J

, ℓ

v+1

), (

^Iv+1

, ℓ

v+2

), . . . , (

^Ik−1

, ℓ

k

), (

^Ik

, ℓ

k

+

1

),

whereJ

= [

minI_v

, π

i

−

1

]

;

– If

λ

i

=

2 (or equivalently, minI_v

= π

i

<

^maxIv), then

U_i

(π) = (

^I1

, ℓ

1

), . . . , (

^Iv−1

, ℓ

v−1

), (

^J

, ℓ

v

), (

^Iv+1

, ℓ

v+1

), . . . , (

^Ik−1

, ℓ

k−1

), (

^Ik

, ℓ

k

+

1

),

whereJ

= [ π

i

+

1

,

^maxIv

]

;

– If

λ

i

=

3 (or equivalently, minI_v

=

maxI_v

= π

i), then

U_i

(π) = (

^I1

, ℓ

1

), . . . , (

^Iv−1

, ℓ

v−1

), (

^Iv+1

, ℓ

v+1

), . . . , (

^Ik−1

, ℓ

k−1

), (

^Ik

, ℓ

k

+

1

).

Example 2. For the permutation

π =

6 2 5 8 7 3 1 4 in Fig. 2,

λ

1

(π), λ

2

(π), . . . , λ

8

(π) =

0

,

⁰

,

¹

,

¹

,

³

,

²

,

¹

,

^{3, and the} process described inDefinition 1gives the slices below.

Remark 1. LetU_i

(π) = (

^I1

, ℓ

1

), (

^I2

, ℓ

2

), . . . , (

^Ik

, ℓ

k

)

^{be the}ith slice of

π

^{, 0}

≤

i

<

n. Then, the following properties can be easily checked:

Fig. 2. The permutationπ=6 2 5 8 7 3 1 4 withb(π)=0 1 1 0 2 3 6 3 andλ1(π), λ2(π), . . . , λ8(π)=0,⁰,¹,¹,³,²,¹,^3.

– the intervalsI₁

,

^I2

, . . . ,

^Ikare in decreasing order, that is maxI_j+1

<

^minIjfor anyj, 1

≤

j

<

^k;

– the sequence

ℓ

1

, ℓ

2

, . . . , ℓ

kis increasing, that is

ℓ

j

< ℓ

j+1for anyj, 1

≤

j

<

^k;

–

{ ℓ

1

, ℓ

2

, . . . , ℓ

k

} ⊆ [

0

,

ⁱ

]

, and

ℓ

k

=

i;

– 0

∈

I_k;

–

∪

^k_j=1I_j

= { π

i+1

, π

i+2

, . . . , π

n

} ∪ {

0

}

;

– the

(

ⁱ

+

1

)

th entry of the Lehmer code of

π

is given by the number of entries

π

j

> π

i+1, withj

<

ⁱ

+

1, that is the cardinality of

[ π

i+1

,

ⁿ

] \ ∪

^k_j=1I_j.

A byproduct ofDefinition 1is the construction ofb:S_n

→

S_ndefined below.

Definition 2. Let

π = π

1

π

2

. . . π

n

∈

S_n. For eachi, 1

≤

i

≤

n, we defineb_i

= ℓ

v, where

v

is such that

(

^Iv

, ℓ

v

)

is a labeled interval in the

(

ⁱ

−

1

)

th slice of

π

^with

π

i

∈

I_v, and we denote byb

(π)

the sequenceb₁b₂

. . .

^bn.

FromRemark 1it follows thatb

(π)

is a subexcedant sequence, see for instanceExample 2andFig. 2.

Proposition 1. Let

π = π

1

π

2

. . . π

n

∈

S_n, b

(π) =

b₁b₂

. . .

^bn, and let an i,1

≤

i

≤

n.

1. i is a descent in

π

iff i is an ascent in b

(π)

^;

2. i is an ides in

π

^{iff b}idoes not occur in b_i+1b_i+2

. . .

^bn; 3. i is a left-to-right maximum in

π

^{iff b}i

=

0;

4. i is a left-to-right minimum in

π

^{iff b}i

=

i

−

1;

5. i is a right-to-left maximum in

π

iff i is right-to-left minimum in b.

Proof. Points 1 and 2 obviously follow from the definition ofb.

Point 3. Letjbe such that

π

j

=

n;b_i

=

0 iffi

≤

jand

π

ilies in the first interval of the

(

ⁱ

−

1

)

th slice of

π

, which in turn is equivalent toiis a left-to-right maximum in

π

^.

Point 4. Similarly, letjbe such that

π

j

=

1;b_i

=

i

−

1 iffi

≤

jand

π

ilies in the last interval of the

(

ⁱ

−

1

)

th slice of

π

^, which in turn is equivalent toiis a left-to-right minimum in

π

^.

Point 5. By the construction ofb,iis a right-to-left maximum in

π

^iff

π

iis the largest element of the first interval of the

(

ⁱ

−

₁

)

th slice of

π

, which in turn is equivalent tob_iis smaller than any ofb_i+1

,

^bi+2

, . . . ,

^bn.

See for instanceExample 1, wheres

=

b

(π)

^.

For a length-nsubexcedant sequenceb

=

b₁b₂

. . .

^bn, let us consider the following properties that a positioni, 1

≤

i

≤

n, can satisfy:

R1: b_ioccurs in the suffixb_i+₁b_i+₂

. . .

^bnofb, R2: i

−

1 occurs inb.

The next proposition shows that each

λ

i

(π)

can be obtained solely fromb

(π)

^. Proposition 2. Let

π ∈

S_nand b

(π) =

b₁b₂

. . .

^bn. Then, for any i,1

≤

i

≤

n, we have

λ

i

(π) =







0

,

if i satisfies both R1 and R2

,

1

,

if i satisfies R2 but not R1

,

2

,

if i satisfies R1 but not R2

,

3

,

if i satisfies neither R1 nor R2

.

Proof. By the construction given inDefinition 2forb

(π)

from the slices of

π

, it follows that the positioniin

π

^satisfies property P1 (resp. P2) if and only if the positioniinb

(π)

satisfies property R1 (resp. R2), and the statement holds.

Proposition 3. Let

π, σ ∈

S_nwith b

(π) =

b

(σ )

^{. Then} 1.

λ

i

(π) = λ

i

(σ )

^{for any i,1}

≤

i

≤

n.

2. If

(

^I1

, ℓ

1

), (

^I2

, ℓ

2

), . . . , (

^Ik

, ℓ

k

)

is the ith slice of

π

^{, and}

(

^J1

,

^m1

), (

^J2

,

^m2

), . . . , (

^Jp

,

^mp

)

^{that of}

σ

, for some i,1

≤

i

<

^{n, then} k

=

p and

ℓ

j

=

m_j, for1

≤

j

≤

k.

Proof. The first point is a consequence ofProposition 2.

The second point follows by the next considerations. Theith slice of

π

^,i

≤

1

<

n, has the same number of intervals as its

(

ⁱ

−

1

)

th slice, except in two cases:

λ

i

(π) =

0 (when an interval is split into two intervals); and when

λ

i

(π) =

3 (when a one-element interval is removed). The result follows by considering the first point and by induction oni.

The sequenceb

(π) =

b₁b₂

. . .

^bn

∈

S_nwas defined by means of the slices of

π

, but in proving the bijectivity ofbwe need rather the complement of these slices. Let

π ∈

S_nandU_i

(π) = (

^I1

, ℓ

1

), (

^I2

, ℓ

2

), . . . , (

^Ik

, ℓ

k

)

^{be the}ith slice of

π

for ani, 1

≤

i

<

^{n. Theithprofileof}

π

is the sequenceX₁

,

^X2

, . . . ,

^Xpof decreasing nonempty maximal intervals (that is, maxX_j+1

<

^minXj, and none of them has the formX_j

∪

X_j+1) with

∪

^p_j=1X_j

= {

1

,

²

, . . . ,

ⁿ

} \ ∪

^k_j=1I_j. And, clearly,

∪

^p_j=1X_jis the set of entries in

π

to the left of

π

i+1, andp

j=1cardX_j

=

i.

Example 3. The vertical gray regions on the right side ofExample 2correspond to the profiles of

π =

6 2 5 8 7 3 1 4 inFig. 2.

These profiles are

[

6

,

⁶

]

;

[

6

,

⁶

] , [

2

,

²

]

;

[

5

,

⁶

] , [

2

,

²

]

;

[

8

,

⁸

] , [

5

,

⁶

] , [

2

,

²

]

;

[

5

,

⁸

] , [

2

,

²

]

;

[

5

,

⁸

] , [

2

,

³

]

; and

[

5

,

⁸

] , [

1

,

³

]

. In the proof ofTheorem 1, we need the next result.

Proposition 4. Let

π, σ ∈

S_nwith b

(π) =

b

(σ)

, and let an i,1

≤

i

<

^{n. If X}1

,

^X2

, . . . ,

^Xpand Y₁

,

^Y2

, . . . ,

^Ymare the ith profiles of

π

^{and of}

σ

^{, then}

– n

∈

X₁if and only if n

∈

Y₁, – p

=

m, and

– cardX_j

=

cardY_jfor any j,1

≤

j

≤

p.

Proof. It is easy to see thatn

∈

X₁iff 0 does not appear inb_i+1

(π) . . .

^bn

(π) =

b_i+1

(σ) . . .

^bn

(σ)

, that is, iffn

∈

Y₁. And, if i

=

1, then the first profile of

π

^{and of}

σ

are one-element intervals, and the statement holds.

From the first point ofProposition 3, we have

λ

i

(π) = λ

i

(σ )

. Suppose that the statement is true fori

−

1, and we will prove it fori.

In passing from the

(

ⁱ

−

1

)

th profiles of

π

^{and of}

σ

^{to their}ith profiles, the following cases can occur (we refer the reader toDefinition 1andFig. 1).

– If

λ

i

(π) = λ

i

(σ ) =

0, or

λ

i

(π) = λ

i

(σ ) =

1 andb_i

(π) =

b_i

(σ ) =

0, then a new one-element interval is added to theith profile of

π

^{and of}

σ

. Moreover, sinceb

(π) =

b

(σ)

, by the second point ofProposition 3, it follows that these intervals are both, for somek, thekth intervals in theith profile of

π

^and

σ

^.

– If

λ

i

(π) = λ

i

(σ ) =

1 andb_i

(π) =

b_i

(σ ) ̸=

0, then for somek, a new element is added to thekth interval of bothith profiles of

π

^and

σ

; this element is the smallest one in the obtained intervals.

– If

λ

i

(π) = λ

i

(σ ) =

2, or

λ

i

(π) = λ

i

(σ) =

3 andb_i

(π) =

b_i

(σ ) =

0, then for somek, a new element is added to thekth interval of bothith profiles of

π

^and

σ

; this element is the largest one in the obtained intervals.

– If

λ

i

(π) = λ

i

(σ ) =

3 andb_i

(π) =

b_i

(σ ) ̸=

0, then two consecutive intervals are merged in theith profiles of

π

^{and of}

σ

^{: thekth and}

(

^k

+

1

)

th ones, for somek.

Now, we explain how the Lehmer codec₁c₂

. . .

^cnis linked to the profiles of a permutation. By definition,c₁

=

0 and c_i,i

>

1, is the number of entries in

π

at the left of

π

iand larger than

π

i. IfX₁

,

^X2

, . . . ,

^Xpis the

(

ⁱ

−

1

)

th profile of

π

^{, it} follows thatc_i

=

u

j=1cardX_j, whereuis such that

∪

^u_j=1X_jis the set of entries in

π

at the left of

π

iand larger than

π

i, and soc_i

=

card

∪

^u_j=1X_j.

Theorem 1. The mapping b:S_n

→

S_nis a bijection.

Proof. Let

π, σ ∈

S_nwithb

(π) =

b

(σ)

^{, andc}1c₂

. . .

^cnandd₁d₂

. . .

^dnbe the Lehmer codes of

π

^and

σ

^{. Let alsoibe an} integer, 1

<

ⁱ

≤

n, and

(

^I1

, ℓ

1

), (

^I2

, ℓ

2

), . . . , (

^Ik

, ℓ

k

)

^{be the}

(

ⁱ

−

1

)

th slice of

π

^{, and}

v

^{such that}

π

i

∈

I_v(seeDefinition 2). If X₁

,

^X2

, . . . ,

^Xpis the

(

ⁱ

−

1

)

th profile of

π

^{, then}

ifn

∈

X₁, it follows thatc_i

=

_v

j=1cardX_j, and ifn

̸∈

X₁, it follows thatc_i

=

_v−1

j=1 cardX_j.

Sinceb

(π) =

b

(σ )

, combiningProposition 4and the second point ofProposition 3, we have thatc_i

=

d_i. It follows that the Lehmer code of

π

^{and of}

σ

are equal, and so are

π

^and

σ

^{, and thusb}is injective. And, by cardinality reasons, it follows that bis bijective.

It is straightforward to see that the 4-tuple of statistics

(

^Des

,

^Lrmax

,

^Lrmin

,

^Rlmax

)

^onSnhas the same distribution as

(

^Asc

,

^Posz

,

^Max

,

^Rlmin

)

^onSn. Indeed, for the Lehmer codeL

(π)

of a permutation

π

^{, we have}

(

^Des

,

^Lrmax

,

^Lrmin

,

^Rlmax

) π = (

^Asc

,

^Posz

,

^Max

,

^Rlmin

)

^L

(π)

^{, see} Property 1. But, generally,Ides

π

is different from RowL

(π)

. For example, if

π =

6 2 5 8 7 3 1 4, thenL

(π) =

0 1 1 0 1 4 6 4,Ides

π = {

3

,

⁵

,

⁷

,

⁸

}

andRowL

(π) = {

5

,

⁷

,

⁸

}

.

CombiningTheorem 1andProposition 1, it follows thatbbehaves not only as the Lehmer code for the above 4-tuples of statistics, but also it transformsIdes

π

^intoRowb

(π)

. Formally, we have the next theorem, which subsequently givesRow as a set-valued partner forAsc, thereby answering to an open question stated in [1].

Theorem 2.For any

π ∈

S_n,

(

^Des

,

^Ides

,

^Lrmax

,

^Lrmin

,

^Rlmax

) π = (

^Asc

,

^Row

,

^Posz

,

^Max

,

^Rlmin

)

^b

(π),

and so the multistatistic

(

^Des

,

^Ides

,

^Lrmax

,

^Lrmin

,

^Rlmax

)

^onSnhas the same distribution as

(

^Asc

,

^Row

,

^Posz

,

^Max

,

^Rlmin

)

^on S_n.

The next corollaries are consequences ofTheorem 2. The first of them is Foata’s result in [4] saying that

(

^asc

,

^row

)

^on subexcedant sequences is a double Eulerian bistatistic.

Corollary 1. The bistatistics

(

^asc

,

^row

)

on the set of subexcedant sequences has the same distribution as

(

^des

,

^ides

)

on the set of permutations.

Corollary 2. The bistatistics

(

^asc

,

^row

)

^and

(

^row

,

^asc

)

are equidistributed on the set of subexcedant sequences.

Proof. Lets

∈

_Snand let us definet

=

_b

(σ )

^where

σ = π

^−1with

π =

_b⁻¹

(

^s

)

. It is clear that

(

^asc

,

^row

)

^s

= (

^des

,

^ides

) π = (

^ides

,

^des

) σ = (

^row

,

^asc

)

^t.

Final remarks: Recently, Lin and Kim [6] defined a bijectionΨ between two combinatorial classes counted by the large Schröder numbers, namely the setS_n

(

⁰²¹

)

of subexcedant sequences avoiding the pattern 021, and the setS_n

(

²⁴¹³

,

⁴²¹³

)

of permutations avoiding the patterns 2413 and 4213. In [6], it is shown thatΨ proves the equidistribution of two 6-tuples of set-valued statistics, that is

(

^Asc

,

^Row

,

^Posz

,

^Max

,

^Rlmin

,

^Expo

)

^s

= (

^Des

,

^Ides

,

^Lrmax

,

^Lrmin

,

^Rlmax

,

^Rlmin

)

Ψ

(

^s

),

for anys

∈

_Sn

(

⁰²¹

)

^{. Here,Expois the}exposed positions statisticdefine in [6], and if the last statistic of both 6-tuples is dropped out, then the 5-tuples in ourTheorem 2are obtained. Also in [6] is pointed out that the restriction of our mapping byields a bijection betweenS_n

(

²⁴¹³

,

⁴²¹³

)

^andSn

(

⁰²¹

)

, but unlikeΨ⁻¹, this restriction ofbdoes not transformRlmin toExpo.

References

[1] E. Aas, The double Eulerian polynomial and inversion tables,arXiv:1401.5653, 2014.

[2] M. Beck, S. Robins, Computing the Continuous Discretely: Integer-Point Enumeration in Polyhedra, in: Undergraduate Texts in Mathematics, Springer, New York, 2007.

[3] D. Dumont, Interprétations combinatoires des nombres de Genocchi, Duke Math. J. 41 (2) (1974) 305–318.

[4] D. Foata, Distributions Eulériennes et Mahoniennes sur le groupe des permutations, in: Higher Combinatorics, in: NATO Adv. Study Inst. Ser., Ser. C:

Math. Phys. Sci., vol. 31, Reidel, Dordrecht, Boston, Mass, 1977, pp. 27–49. (in French). With a comment by Richard P. Stanley.

[5] D.H. Lehmer, Teaching combinatorial tricks to a computer, in: Proc. Sympos. Appl. Math., Vol. 10, Amer. Math. Soc., 1960, pp. 179–193.

[6] Z. Lin, D. Kim, A sextuple equidistribution arising in pattern avoidance,arXiv:1612.02964v2, 2016.

[7] R. Mantaci, F. Rakotondrajao, A permutation representation that knows what Eulerian means, Discrete Math. Theor. Comput. Sci. 4 (2) (2001) 101–108 (electronic).

[8] T.K. Petersen, Two-sided Eulerian numbers via balls in boxes, Math. Mag. 86 (2013) 159–176.

[9] C.D. Savage, M.J. Schuster, Ehrhart series of lecture hall polytopes and Eulerian polynomials for inversion sequences, J. Combin. Theory Ser. A 119 (4) (2012) 850–870.

[10]M. Skandera, Dumont’s statistic on words, Electron. J. Combin. 8 (1) (2001) #R11.

[11]V. Vajnovszki, Lehmer code transforms and Mahonian statistics on permutations, Discrete Math. 313 (2013) 581–589.