BULLETIN OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 80, Number 5, September 1974
ENUMERATION OF PAIRS OF PERMUTATIONS
AND SEQUENCES1
BY L. CARLITZ, RICHARD SCOVILLE AND THERESA VAUGHAN
Communicated by Paul T. Bateman, January 10, 1974
Let ir=(al9 • • • , an) denote a permutation of Z W ={1, 2, • • • , n).
A rise of TT is a pair ai9 ai+1 with ai<ai+1; & fall is a pair ai9 ai+1 with
cti>ai+1. Thus if p = (bl9 • • • , bn) denotes another permutation of Zn9
the two pairs ai9 ai+1; bi9 bi+1 are either both rises, both falls, a rise and a
fall or a fall and a rise. We denote these four possibilities by RR9 FF9 RF9
FR9 respectively.
Let a>(n) denote the number of pairs of permutations n9 p with RR
forbidden. More generally let œ(n9 k) denote the number of pairs ir9 p
with exactly k occurrences of RR.
THEOREM
1.
We have
oo
„n
JÎ3D
n\n\
too
„n
\— 1
l^o
nlnl)
where co(0)=co(l) = l.
THEOREM 2.
oo
_n
n-1
-i _
n
Y
y -^— y < > ®x* = — - — - — >
(2)
wAere/(z)=2SU ( - I f (^//I!/I!).
The pair TT9 p is said to be amicable if i?F and FR are both forbidden.
Let <x(n) denote the number of amicable pairs of Zn; more generally let
a(n9 k) denote the number of pairs TT9 p with k total occurrences of RF
and FR.
THEOREM
(3)
3.
We have
A(z)A(-z) = 1,
wAere ^(z)=2^Lo (x.(n)znjn\n\.
AMS (MOS) subject classifications (1970). Primary 05A15, 10A45.
Key words and phrases. Permutations, sequences, generating functions.
1
Supported in part by NSF grant GP-37924.
Copyright © American Mathematical Society 1974
881
882
L. CARLITZ, R. SCOVILLE AND T. VAUGHAN
[September
Equation (3) is equivalent to a ( 0 ) = l and
(4)
V (-lWffa(fc)a(n - k) = 0
(n > 0).
Unfortunately (4) does not suffice to determine oi(ri).
THEOREM
4.
We have
Ùxn\n\£«
l-yA(x(l-y))
We next consider pairs of sequences. The sequence o=(al9 #2> • • • , aN)
{ai e Z J is said to be of specification [e]=[e 1 , • • • , en] if each element
in Zn occurs exactly ej times, where e± + - • -+en=N. Enumeration of
sequences subject to various requirements has been discussed in a number
of papers [1], [2], [3], [4], [5]. The pair ai9 ai+1 is a rise, fall or level
according as at<ai+l9 ai>ai+l9 a~ai+1 ( / = 1 , 2, • • • , N-l).
Let r=(b1, - - - , bN) denote a sequence of specification [ƒ] =
L/i> ' * ' > fn\- Then for the pair a, T there are now nine possibilities,
namely
(6)
RR. FR, LR, RF, FF, LF, RL, FL, LL.
{n)
Let Q (r; e,f) denote the number of pairs of sequences a, r of specification [e], [ƒ], respectively, and with exactly N—r—l occurrences of
RR and put
Q{n\x, y,z)
=^Q{n\r;e,f)xyz\
e
>f>r
x
where x*=x{ •
THEOREM
5.
• • xenn.
We have
Q{n\x,y,z)
(7)
=
llDn,
where
Dn = 1 - S1(x)S1(y) + (1 - z)S,(x)S2(y)
+ (-1)"(1 -
zy-'S^Sniy)
and Sk(x) is the kth elementary symmetric function of xl9 • • • , xn. In
particular the generating function for pairs of sequences with RR forbidden
is
(8)
{1 - S^S^y)
+ S2(x)S2(y)
+ (-lrs^x)^}"1.
Let M{n)(r; e,f) denote the number of pairs a, r of specification [e],
[ƒ] with exactly N—r—l occurrences of LL and put
M{n\x, y, z) = J M{n\r;
e
J,r
e,f)xeyf.
1974]
PAIRS OF PERMUTATIONS AND SEQUENCES
THEOREM
6.
We have
M
(9)
883
M (x,y,z)
=
( l - z ) i( l +faf
l
(1 Z)X
-
^ I
1 - (1 - z)x^,j
OU 1 - (1 - z)xiyi.
In particular the generating function for pairs with LL forbidden is
(10)
(l-f
I
-^i-)"1.
i,^=i 1 +
XJV
Let y4? B denote any disjoint partition A?£0, J 5 ^ 0 of the set (6).
Let C(e,f k) denote the number of pairs of sequences a, r with exactly
k JB'S. Put
^ ( * , y) = 2 C(e, ƒ, fc)jcV (fc = 0, 1, 2, • • •),
where C(e,f9 0) = 1 (tf=0, 1); F(x, y9 z) =
THEOREM
(11)
7.
2Z.0z*Fk(x,y).
fFe have
F(x
z) = (1 - z ) f o(d - *)*, y) = (1 - z)F0(x, (1 - z)y)
**
1 - zF 0 ((l - z)x, j;)
1 - zF 0 (x, (1 - z)y) *
THEOREM 8. Le/ CA(e,f) denote the number of pairs a, r with A forbidden. Then
(i2)
2 c>>/>v =
t?
(13)
FA(x9 y)FB(-x,
* , = vr—, •
F 0 (-Jf,j?)
F 0 (*, - ƒ )
y) = FA(x9 y)FB{x9 -y) = 1,
where FA(x9 y) denotes the left member of (12).
A fuller account of these and other results will appear elsewhere.
REFERENCES
1. L. Carlitz, Enumeration of sequences by rises and falls: A refinement of the Simon
Newcomb problem, Duke Math. J. 39 (1972), 267-280. MR 45 #1778.
2.
, Enumeration of up-down sequences, Discrete Math. 4 (1973)> 273-286.
MR 47 #1625.
3. L. Carlitz and Richard Scoville, Up-down sequences, Duke Math. J. 39 (1972),
583-598. MR 47 #1627.
884
L. CARLITZ, R. SCOVILLE AND T. VAUGHAN
4. L. Carlitz, Richard Scoville and Theresa Vaughan, Enumeration of permutations
and sequences with restrictions, Duke Math. J. 40 (1973), 723-741.
5. J. F . Dillon and D . P. Roselle, Simon Newcomb''sproblem, SIAM J. Appl. Math.
17 (1969), 1086-1093. M R 41 #1553.
DEPARTMENT OF MATHEMATICS, D U K E UNIVERSITY, DURHAM, N O R T H
CAROLINA
27706
Current address (Vaughan): Department of Mathematics, University of North
Carolina, Greensboro, North Carolina 27412