INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 8 (2008), #A24
A NOTE ON BOOLEAN LATTICES AND FAREY SEQUENCES II
Andrey O. Matveev
Data-Center Co., RU-620034, Ekaterinburg, P.O. Box 5, Russian Federation
aomatveev@dc.ru, aomatveev@hotmail.com
Received: 2/1/08, Revised: 4/1/08, Accepted: 5/2/08, Published: 5/21/08
Abstract
We establish monotone bijections between subsequences of the Farey sequences and the halfsequences of Farey subsequences associated with elements of the Boolean lattices.
1. Introduction
The Farey sequence of order n, denoted by Fn , is the ascending sequence of irreducible
fractions hk ∈ Q such that 01 ≤ hk ≤ 11 and 1 ≤ k ≤ n; see, e.g., [3, Chapter 27], [4, §3],
[5, Chapter 4], [6, Chapter III], [11, Chapter 6], [12, Chapter 6], [13, Sequences A006842
and A006843], [14, Chapter 5].
The Farey sequence of order n contains the subsequences
!
"
Fnm := hk ∈ Fn : h ≤ m ,
(1)
for all integers m ≥ 1, and the subsequences
Gnm :=
!h
k
"
∈ Fn : k − h ≤ n − m ,
(2)
for m ≤ n − 1, that inherit many familiar properties of the Farey sequences. In particular,
if n > 1 and m ≥ n − 1, then Fnm = Fn ; if n > 1 and m ≤ 1, then Gnm = Fn .
The Farey subsequence Fnm was presented in [1], and some comments were given in [10,
Remark 7.10].
Let B(n) denote the Boolean lattice of rank n > 1, whose operation of meet is denoted
by ∧. If a is an element of B(n), of rank ρ(a) =: m such that 0 < m < n, then the integers n
and m determine the subsequence
#
$
%
!
"
ρ(b∧a)
ρ(b)
F B(n), m : = gcd(ρ(b∧a),ρ(b))
:
b
∈
B(n),
ρ(b)
>
0
gcd(ρ(b∧a),ρ(b))
(3)
!h
"
= k ∈ Fn : h ≤ m, k − h ≤ n − m
of Fn , considered in [7, 8, 9, 10].
INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 8 (2008), #A24
Notice that the map
!
"
!
"
F B(n), m → F B(n), n − m ,
h
k
'→
k−h
k
,
1
h
[ hk ] '→ [ −1
0 1 ] · [k ] ,
2
(4)
is order-reversing and bijective, by analogy with the map
Fn → Fn ,
h
k
'→
k−h
k
1
h
[ hk ] '→ [ −1
0 1 ] · [k ] ,
,
(5)
which is order-reversing and bijective as well; here [ hk ] ∈ Z2 is a vector presentation of
the fraction hk .
We call the ascending sets
"
!
!
"
1!
F ≤ 2 B(n), m := hk ∈ F B(n), m :
and
h
k
≤
1
2
"
,
"
!
!
"
"
1!
F ≥ 2 B(n), m := hk ∈ F B(n), m : hk ≥ 12 .
!
"
the left and right halfsequences of the sequence F B(n), m , respectively.
!
"
This work is a sequel to note [7] which concerns the sequences F B(2m), m . See [7] for
more about Boolean lattices, Farey (sub)sequences and related combinatorial identities.
The key observation is Theorem 3 which in particular! asserts "that there is a bijection
m
between the sequence Fn−m
and the left halfsequence of F B(n), m , on the one
! hand; there
"
n−m
is also a bijection between the sequence Fm and the right halfsequence of F B(n), m , on
the other hand.
Throughout the note, n and m represent integers; n is always greater than one. The
fractions of Farey (sub)sequences are indexed starting with zero. Terms ‘precedes’ and
‘succeeds’ related to pairs of fractions!always mean
relations of immediate consecution. When
"
we deal with a Farey subsequence F B(n), m , we implicitly suppose 0 < m < n.
2. The Farey Subsequences Fnm and Gnm
The connection between the sequences of the form (1) and (2) comes from their definitions:
Lemma 1 The maps
Fnm → Gnn−m ,
h
k
'→
k−h
k
,
1
h
[ hk ] '→ [ −1
0 1 ] · [k ] ,
Gnm → Fnn−m ,
h
k
'→
k−h
k
,
1
h
[ hk ] '→ [ −1
0 1 ] · [k ] ,
and
are order-reversing and bijective, for any m, 0 ≤ m ≤ n.
INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 8 (2008), #A24
3
Following [2, §4.15], we let µ(·) denote the Möbius function on positive integers. For
an interval of positive integers [i, l] := {j : i ≤ j ≤ l} and for a positive integer h, let
φ(h; [i, l]) := |{j ∈ [i, l] : gcd(h, j) = 1}|.
We now describe several basic properties of the sequences Gnm which are dual, in view
of Lemma 1, to those of the sequences Fnn−m , cf. [10, Remark 7.10].
Remark 2 Suppose 0 ≤ m < n.
(i) In Gnm = (g0 < g1 < · · · < g|Gnm |−2 < g|Gnm |−1 ), we have
0
1
g0 =
,
g1 =
1
min{n−m+1,n}
,
g|Gnm |−2 =
n−1
n
,
g|Gnm |−1 =
1
1
.
(ii) (a) The cardinality of the sequence Gnm equals
1+
&
j∈[1,n]
!
"
φ j; [max{1, j + m − n}, j]
&
=1+
j∈[1,n−m+1]
!
"
φ j; [1, j] +
=1+
&
j∈[n−m+2,n]
&
µ(d)·
d≥1
!
"
φ j; [j + m − n, j]
#' (
n
d
' n−m (% ' n−m
(
−
·
+
1
.
d
d
1
2
(b) If gt ∈ Gnm − { 01 }, then
t=
&
j∈[1,n]
=
!
"
φ j; [max{1, j + m − n}, (j · gt )]
&
j∈[1,n−m+1]
!
"
φ j; [1, (j · gt )] +
&
j∈[n−m+2,n]
!
"
φ j; [j + m − n, (j · gt )]
)
&
' n−m ( #' n ( 1 ' n−m
(%
=1+
µ(d)·
· d − 2 d +1
d
d≥1
−
(iii) Let
h
k
∈ Gnm ,
0
1
<
h
k
&
j∈[1,&n/d']
,
*'
+
(
min n−m
, (j ·(1 − gt ))
.
d
< 11 .
(a) Let x0 be the integer such that kx0 ≡ −1 (mod h) and
h + 1 ≤ x0 ≤ .(
m.
' m−
- n−m+x
kx0 +1
∗
∗
0 −y0 n−y0
Define integers y0 and t by y0 := h and t := min
, k
.
k−h
x0 +t∗ h
h
m
The fraction y0 +t∗ k precedes the fraction k in Gn .
(b) Let x0 be the integer such that kx0 ≡ 1 (mod h) and' m −- h + 1 ≤ x0 ≤ .(
m.
n−m+x0 −y0 n−y0
kx0 −1
∗
∗
Define integers y0 and t by y0 := h and t := min
, k
.
k−h
x0 +t∗ h
h
m
The fraction y0 +t∗ k succeeds the fraction k in Gn .
4
INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 8 (2008), #A24
(iv) (a) If
hj
kj
<
hj+1
kj+1
are two successive fractions of Gnm then
kj hj+1 − hj kj+1 = 1 .
(b) If
hj
kj
<
hj+1
kj+1
<
hj+2
kj+2
are three successive fractions of Gnm then
hj+1
kj+1
h
h
=
hj +hj+2
gcd(hj +hj+2 ,kj +kj+2 )
$
kj +kj+2
gcd(hj +hj+2 ,kj +kj+2 )
.
h
j+1
j+2
(c) If kjj < kj+1
< kj+2
are three successive fractions of Gnm then the integers hj , kj ,
hj+2 and kj+2 are related in the following way:
/
*
+0
k
+n kj+2 −hj+2 +n−m
hj = min j+2
,
hj+1 − hj+2 ,
kj+1
kj+1 −hj+1
/
*
+0
k
+n kj+2 −hj+2 +n−m
kj = min j+2
, kj+1 −hj+1
kj+1 − kj+2 ,
kj+1
/
*
+0
k +n k −h +n−m
hj+2 = min kjj+1 , kj j+1j−hj+1
hj+1 − hj ,
/
*
+0
k +n k −h +n−m
kj+2 = min kjj+1 , kj j+1j−hj+1
kj+1 − kj .
(v) If
1
∈ Gnm , where
k
&min{ n−m−1
, n−1
}'
k−1
k
n > 1, for some k > 1, then the fraction
k&min{ n−m−1
, n−1
}'+1
k−1
k
precedes k1 , and the fraction
&min{ n−m+1
, n+1
}'
k−1
k
k&min{ n−m+1
, n+1
}'−1
k−1
k
succeeds
1
k
in Gnm .
!
"
3. The Farey Subsequence F B(n), m
!
"
Definition (3) implies that a Farey subsequence F B(n), m can be regarded as the intersection
!
"
F B(n), m = Fnm ∩ Gnm .
Its halfsequences can be described with the help of the following statement:
!
"
Theorem 3 Consider a Farey subsequence F B(n), m . The maps
and
"
1!
m
F ≤ 2 B(n), m → Fn−m
,
"
1!
m
Fn−m
→ F ≤ 2 B(n), m ,
"
1!
2m−n
F ≥ 2 B(n), m → Gm
,
"
1!
2m−n
Gm
→ F ≥ 2 B(n), m ,
h
k
h
k
h
k
h
k
'→
'→
h
k−h
,
h
k+h
,
1 0
h
[ hk ] '→ [ −1
1 ] · [k ] ,
[ hk ] '→ [ 11 01 ] · [ hk ] ,
(6)
(7)
'→
2h−k
h
,
h
[ hk ] '→ [ 21 −1
0 ] · [k ] ,
(8)
'→
k
2k−h
,
0 1
h
[ hk ] '→ [ −1
2 ] · [k ] ,
(9)
INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 8 (2008), #A24
5
are order-preserving and bijective.
The maps
"
1!
n−2m
F ≤ 2 B(n), m → Gn−m
,
"
1!
n−2m
Gn−m
→ F ≤ 2 B(n), m ,
"
1!
n−m
F ≥ 2 B(n), m → Fm
,
h
k
h
k
h
k
'→
'→
k−2h
k−h
,
k−h
2k−h
,
[ hk ] '→
[ hk ] '→
1 −2 1 2
−1 1
1 −1 1 2
−1 2
· [ hk ] ,
· [ hk ] ,
'→
k−h
h
,
1
h
[ hk ] '→ [ −1
1 0 ] · [k ] ,
(10)
'→
k
k+h
,
[ hk ] '→ [ 01 11 ] · [ hk ] ,
(11)
and
"
1!
n−m
Fm
→ F ≥ 2 B(n), m ,
h
k
are order-reversing and bijective.
Proof. For any integer h, 1 ≤ h ≤ m, we have
3- h
!
"
.3
3 ∈ F B(n), m : h < 1 3 = φ(h; [2h + 1, h + n − m])
k
2
& k
&
!'
( 2h "
!' n−m ( h "
=
µ(d) · h+n−m
−
=
µ(d)
·
−d
d
d
d
d∈[1,h]: d|h
d∈[1,h]: d|h
3.3
m
= φ(h; [h + 1, n − m]) = 3 hk ∈ Fn−m
: hk < 11 3
"
1!
m
and see that the sequences F ≤ 2 B(n), m and Fn−m
are of the same cardinality, but this
3 ≥1 !
"3
n−m
conclusion also implies 3F 2 B(n), m 3 = |Fm |, due to bijection (4). The proof of the
assertions concerning maps (6), (7), (10) and (11) is completed by checking that, on the
h
hj+1
hj
m
one hand, a fraction kjj precedes a fraction kj+1
in Fn−m
if and only if the fraction kj +h
j
!
"
hj+1
hj
≤ 12
precedes the fraction kj+1 +hj+1 in F
B(n), m ; on the other hand, a fraction kj precedes a
!
"
hj+1
kj+1
kj
≥ 12
n−m
fraction kj+1 in Fm if and only if kj+1 +hj+1 precedes kj +h
in
F
B(n),
m
. The remaining
j
assertions of the theorem now follow, thanks to Lemma 1.
!
For example,
!
F6 = 01 <
!
F64 = 01 <
!
G64 = 01
!
" !
F B(6), 4 = 01
!
4
F6−4
= F2 = 01
G42·4−6 = G42 =
!
6−2·4
G6−4
= F2 = 01
F46−4 = F42 =
1
6
1
6
<
<
1
5
1
5
<
<
1
4
1
4
<
<
<
<
<
1
3
1
3
1
3
1
3
1
2
<
1
2
<
<
2
5
2
5
"
< 12 < 35 < 23 < 34 < 45 < 56 < 11 ,
"
< 12 < 35 < 23 < 34 < 45
< 11 ,
"
< 12 < 35 < 23 < 34 < 45 < 56 < 11 ,
"
< 12 < 35 < 23 < 34 < 45
< 11 ,
"
< 11 ,
!0
"
1
1
2
3
1
<
<
<
<
<
,
1
2
3
4
1
" 3
1
<1 ,
!0
"
1
1
1
2
1
<
<
<
<
<
.
1
4
3
2
3
1
Theorem 3 allows us to write down the formula
3 !
3 min{m,n−m} 3 3 min{m,n−m} 3
"3
m
n−m
3F B(n), m 3 = |Fn−m
3 + 3Fm
3−1 ,
| + |Fm
| − 1 = 3Fn−m
6
INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 8 (2008), #A24
!
"
for the number
of
elements
of
a
sequence
F
B(n),
m
, cf. [10, Proposition 7.3(ii)]. Recall that
!' (
' (" '
(
4
|Fqp | = 1 + d≥1 µ(d) · dq − 12 dp · dp + 1 , for any Farey subsequence Fqp with 0 < p ≤ q;
' (
4
see [10, Remark 7.10(ii)(b)]. Notice that the well-known relation d≥1 µ(d)· dt = 1, for any
' ( !' (
' ("
4
positive integer t, leads us to one more formula: |Fqp | = 32 + d≥1 µ(d) · dp · dq − 12 dp .
Thus, we have
or
/
0 #'
&
3 !
"3
( 1 / min{m,n−m} 0%
n−m
3F B(n), m 3 = 3 +
µ(d)· min{m,n−m}
·
−2
d
d
d
2 d≥1
/
0 #' (
/
0%
3 &
m
1 min{m,n−m}
+ +
µ(d)· min{m,n−m}
·
−
−1
d
d
2
d
2 d≥1
/
0 #'
&
3 !
"3
( ' m ( / min{m,n−m} 0%
min{m,n−m}
n−m
3F B(n), m 3 = 2 +
µ(d)·
·
+ d −
,
d
d
d
d≥1
that is,
&
3 !
"3
' ('
(
3F B(n), m 3 − 2 =
µ(d)· md · n−m
.
d
d≥1
In particular, for any positive integer t we have
&
' (2 3 !
"3
µ(d)· dt = 3F B(2t), t 3 − 2 = 2 |Ft | − 3 .
d≥1
!
"
Proposition 4 Consider a Farey subsequence F B(n), m .
(i) Suppose m ≥ n2 . The map
"
"
1!
1!
F ≤ 2 B(n), m → F ≤ 2 B(n), m ,
h
k
'→
k−2h
2k−3h
,
[ hk ] '→
1 −2 1 2
−3 2
· [ hk ] ,
h
k
'→
k−h
2k−3h
,
[ hk ] '→
1 −1 1 2
· [ hk ] ,
h
k
'→
k−h
k
is order-reversing and bijective. The map
"
"
1!
1!
F ≤ 2 B(n), m → F ≥ 2 B(n), m ,
is order-preserving and injective. The map
"
"
1!
1!
F ≤ 2 B(n), m → F ≥ 2 B(n), m ,
−3 2
(12)
1
h
[ hk ] '→ [ −1
0 1 ] · [k ] ,
,
is order-reversing and injective.
(ii) Suppose m ≤ n2 . The map
"
"
1!
1!
F ≥ 2 B(n), m → F ≥ 2 B(n), m ,
h
k
'→
h
3h−k
,
0
[ hk ] '→ [ 13 −1
] · [ hk ] ,
(13)
INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 8 (2008), #A24
is order-reversing and bijective. The map
"
"
1!
1!
F ≥ 2 B(n), m → F ≤ 2 B(n), m ,
is order-preserving and injective. The map
"
"
1!
1!
F ≥ 2 B(n), m → F ≤ 2 B(n), m ,
h
k
'→
2h−k
3h−k
h
k
'→
k−h
k
[ hk ] '→
,
1 2 −1 2
3 −1
7
· [ hk ] ,
1
h
[ hk ] '→ [ −1
0 1 ] · [k ] ,
,
is order-reversing and injective.
m
Proof. To prove that map (12) is order-reversing and bijective, notice that Fn−m
= Fn−m ,
and consider the composite map
(6)
1
h
k
F ≤ 2 (B(n),m)−→Fn−m
'→
(7)
(5)
h
k−h
Fn−m −→Fn−m
'→
k−2h
k−h
1
Fn−m −→F ≤ 2 (B(n),m)
'→
k−2h
2k−3h
.
2m−n
Similarly, under the hypothesis of assertion (ii) we have Gm
= Fm ; consider the composite
map
(8)
(9)
(5)
≥1
≥1
h F 2 (B(n),m)−→Fm 2h−k Fm −→Fm k−h Fm −→F 2 (B(n),m)
h
'→
'→
'→
k
h
h
3h−k
to see that map (13) is order-reversing and bijective.
The remaining assertions follow from the observation that map (4) is the order-reversing
bijection.
!
!
"
We conclude the note by listing a few
pairs
of
fractions
adjacent
in
F
B(n),
m
; see [8, 9]
!
"
on such pairs within !the sequences
F
B(2m),
m
.
We
find
the
neighbors
of
the
images of
"
several fractions
of "F B(n), m under bijections from Theorem 3, and we then reflect them
!
back to F B(n), m :
!
"
Remark 5 Consider a Farey subsequence F B(n), m , with n ,= 2m.
(i) Suppose m > n2 .
The fraction
n−m−1
2(n−m)−1
precedes 12 , and the fraction
The fraction
2 min{n−m,& m+1
'}−1
2
3 min{n−m,& m+1
'}−1
2
n−m+1
2(n−m)+1
precedes 23 ; the fraction
succeeds 12 .
2 min{n−m,& m−1
'}+1
2
3 min{n−m,& m−1
'}+1
2
If we additionally have n − m > 1, then the fraction

$
 n−m−2 3(n−m)−4 , if n − m is even,
2
2
$
 n−m−1 3(n−m)−1 , if n − m is odd,
2
2
precedes 13 ; the fraction
succeeds 13 .

$
 n−m 3(n−m)−2 ,
if n − m is even,
2
$ 2
 n−m+1 3(n−m)+1 , if n − m is odd,
2
2
succeeds 23 .
INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 8 (2008), #A24
8
(ii) Suppose m < n2 .
The fraction
m
2m+1
precedes 12 , and the fraction
The fraction
min{m,& n−m+1
'}−1
2
3 min{m,& n−m+1
'}−2
2
m
2m−1
succeeds 12 .
precedes 13 ; the fraction
min{m,& n−m−1
'}+1
2
3 min{m,& n−m−1
'}+2
2
succeeds 13 .
If we additionally have m > 1, then the fraction

$
(m − 1) 3m−2 , if m is even,
2
$
m 3m+1 ,
if m is odd,
2
precedes 23 ; the fraction
succeeds 23 .

$
(m − 1) 3m−4 , if m is even,
2
$
m 3m−1 ,
if m is odd,
2
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