On the Enumeration
of Skew Young Tableaux
Rihard P. Stanley
Department of Mathematis
Massahusetts Institute of Tehnology
Cambridge, MA 02139
e-mail: rstanmath.mit.edu
1
version of 9 September 2001
1
Introdution.
A reent paper [7℄ of MKay, Morse, and Wilf onsiders the number N (n; T )
of standard Young tableaux (SYT) with n ells that ontain a xed standard
Young tableau T of shape ` k. (For notation and terminology related to
symmetri funtions and tableaux, see [6℄ or [12, Ch. 7℄.) They obtain the
asymptoti formula
tn f ;
(1)
N (n; T ) k!
where f denotes the number of SYT of shape and tn denotes the number
of involutions in the symmetri group Sn . Note that N (n; T ) = N (n; U )
whenever T and U are SYT of the same shape. Hene we an write N (n; )
for N (n; T ). Moreover, it is lear that
N (n; ) =
X
`n
f = ;
(2)
where f = denotes the number of SYT T of skew shape =.
In Setion 2 we extend equation (1), using tehniques from the theory of
symmetri funtions, to give an expliit formula for N (n; ) as a nite linear
ombination of tn j 's, from whih in priniple we an write down the entire
asymptoti expansion of N (n; ). In Setion 3 we apply similar tehniques,
1
Partially supported by NSF grant #DMS-9988459.
1
together with asymptoti formulas for harater values of Sn due to Biane
[2℄ and to Vershik and Kerov [14℄, to derive the asymptoti behavior of f =
as a funtion of for xed .
2
A formula for
N (n; ).
Let () denote the value of the irreduible harater of Sk on a permutation of yle type ` k (as explained e.g. in [6, x1.7℄ or [12, xx7.17{7.18℄).
Let mi () denote the number of parts of the partition equal to i, and write
~ for the partition obtained from by replaing every even part 2i with the
two parts i; i. For instane,
= (6; 6; 5; 4; 2; 1)
) ~ = (5; 3; 3; 3; 3; 2; 2; 1; 1; 1):
Equivalently, if w is a permutation of yle type , then w has yle type
~. Note that a permutation of yle type ~ is neessarily even. We will use
notation suh as (~
; 1k j ) to denote a partition whose parts are the parts of
~ with k j additional parts equal to 1. Finally we let z denote the number
of permutations ommuting with a xed permutation of yle type , so
2
z = 1m
2m2 ()
1( )
m ()! m ()! :
1
2
The main result of this setion is the following.
2.1 Theorem.
Let ` k. Then for n k we have
N (n; ) =
k
X
j =0
tn j
(k j )!
X
`
z (~; 1k j ):
1
(3)
j
m1 ()=m2 ()=0
Let ` n k, and let p = p1 p2 denote the power sum
symmetri funtion indexed by . Similarly s= denotes the skew Shur
funtion indexed by =. Sine for any homogeneous symmetri funtion f
of degree n k we have that hpn k ; f i is the oeÆient of x xn k in f ,
Proof.
1
1
2
and sine the oeÆient of x xn k in s= is f = , we have (using a basi
property [6, (5.1)℄[12, Thm. 7.15.4℄ of the standard salar produt h; i on
symmetri funtions)
1
f = = hpn k ; s= i
= hpn k s ; s i:
1
1
Summing on ` n gives
*
N (n; ) =
p1n k s ;
X
`n
+
s :
(4)
Now [6, Exam. I.5.4, p. 76℄[12, Cor. 7.13.8℄
X
s =
Q
i (1
xi ) 1
Q
i<j (1
xi xj )
;
summed over all partitions of all n 0. Sine
Q
i (1
xi ) 1
Q
xi xj )
i<j (1
= exp
X
1 X
n
n
i
1
xni +
X
i<j
!
xni xnj
!
Xp
pn
n ;
= exp
+
2
n
1
2
n
n
n
X
2
1
1
there follows
X
`n
s =
X
X
m
=(1 1
=
`n
;
m
2 2
z pm
1
;:::)`n
1
2
1
m2 p2m4 pm3 +2m6 p2m8
1 +2
2
3
4
z p ;
1
(5)
~
where (1m1 ; 2m2 ; : : :) denotes the partition with mi parts equal to i.
It follows from [6, p. 76℄[12, solution to Exer. 7.35(a)℄ that for any symmetri funtions f and g we have
hp f; gi = f; p g ;
1
1
3
where p 1 g indiates that we are to expand g as a polynomial in the pi 's and
then dierentiate with respet to p . Applying this to equation (4) and using
(5) yields
1
*
N (n; ) =
n k X
s ; n k
z p
p `n 1
*
~
1
X
s ;
=
+
where mi = mi () and (a)n
k
`n
+
z (m + 2m )n k p
1
1
2
1) (a
= a(a
1
n+k p
;
~
(6)
n + k + 1).
Fix m + 2m = n j in equation (6). Thus = (; 2m2 ; 1m1 ) for some
unique ` k satisfying m () = m () = 0. Sine n!=z is the number of
permutations in Sn of yle type , we have for xed ` j that
1
2
1
2
X
=(;2m2 ;1m1 )
Moreover,
p
It follows that
*
N (n; ) =
s ;
*
=
s ;
1
n!
=t
z n
n+k p
~
n j!
:
j z
j
= pk j p :
1
~
k
X
n
j =0
k
X
j =0
t
(n j )n k n j
j!
n!
j
tn j
(k j )!
X
`
+
X
z p
1
`
k j
1
p
~
j
m1 ()=m2 ()=0
+
z p
1
k j
1
p :
~
j
m1 ()=m2 ()=0
Sine [6, (7.7)℄[12, p. 348℄
hs; pk j p i = (~; 1k j );
1
the proof follows.
~
2
Note that the restrition n
N (n; ) = 0 for n < k.
k
in Theorem 2.1 is insigniant sine
4
Theorem 2.1 expresses N (n; ) as a linear ombination of the funtions
tn j , 0 j k. Sine tn j = o(tn j ), this formula for N (n; ) is atually
an asymptoti expansion. The rst few terms are
1
1 1
f tn +
(3; 1k )tn
k!
3(k 3)!
1
1
(2; 2; 1k )tn +
(5; 1k )tn
+
4(k 4)!
5(k 5)!
2
+
(3; 3; 1k )tn + O(tn ):
(7)
9(k 6)!
Note that by symmetry it is lear that if 0 is the onjugate partition to then N (n; ) = N (n; 0 ). Indeed, sine a permutation of yle type ~ is even
we have (~
; 1k j ) = 0 (~; 1k j ). The exat formulas for N (n; ) when
jj 5 and jj n are given as follows (where we write e.g. N (n; 21) for
N (n; (2; 1))):
N (n; ) =
3
3
4
5
4
5
6
6
7
N (n; 1) = tn
N (n; 2) = N (n; 11) =
N (n; 3) = N (n; 111) =
N (n; 21) =
N (n; 4) = N (n; 1111) =
N (n; 31) = N (n; 211) =
N (n; 22) =
N (n; 5) = N (n; 11111) =
N (n; 41) = N (n; 2111) =
N (n; 32) = N (n; 221) =
N (n; 311) =
1
t
2n
1
(t + 2tn )
6 n
1
(t t )
3 n n
1
(t + 8tn + 6tn )
24 n
1
(t 2tn )
8 n
1
(t 4tn + 6tn )
12 n
1
(t + 20tn + 30tn + 24tn )
120 n
1
(t + 5tn
6tn )
30 n
1
(t 4tn + 6tn )
24 n
1
(t 10tn + 4tn ):
20 n
3
3
3
4
3
4
4
3
3
5
3
4
4
5
4
5
5
The omplete asymptoti expansion of tn beginning
tn p1 nn= e
2
n
2
2
pn
+
1
4
119
+
1152n
7
1+ p
24 n
was obtained by Moser and Wyman [8, 3.39℄. In priniple this an be used
to obtain the asymptoti expansion of N (n; ) in terms of more \familiar"
funtions than tn j . The rst few terms an be obtained from the formula
tn
j
1 n j
=p n 2 e
2
n
2
pn
+
1
4
j
7
1+
24
pn
2
+O
1
1
n=
3 2
7
119
+ j
1152 48
3
j
8
2
1
n
;
though we omit the details.
Instead of ounting the number N (n; ) of SYT with n ells ontaining a
xed SYT T of shape , we an ask (as also done in [7℄) for the probability
P (n; ) that a random SYT with n ells (hosen from the uniform distribution on all SYT with n ells) ontains T as a subtableau. Sine the total
number of SYT with n ells is tn , we have
P (n; ) = N (n; )=tn :
Let ej () denote the oeÆient of tn
ej () =
(k
1
j
in the right-hand side of (3), viz.,
X
j )!
z (~; 1k j ):
(8)
1
`
j
m1 ()=m2 ()=0
It follows from Theorem 2.1, using the fat that e () = f =k! and e () =
e () = 0, that
0
1
2
P (n; ) =
f e ( )
+
k! n =
3
3e ( )
3
3 2
n
2
2e ( )
4
+O n
= :
5 2
The leading term of this expansion was obtained in [7, Thm. 1℄.
There is an alternative formula for N (n; ) whih, though not as onvenient for asymptotis, is more ombinatorial than equation (3) beause it
6
avoids using the haraters of Sn. This formula ould be derived diretly
from Theorem 2.1, but we give an alternative proof whih is impliitly bijetive (sine the formulas on whih it is based have bijetive proofs).
Let ` k. Then for all n 0 we have
2.2 Theorem.
N (n + k ; ) =
k X
n
j
j =0
!
X
`k j
f =
tn j :
(9)
We begin with the following Shur funtion identity, proved independently by Lasoux, Madonald, Towber, Stanley, Zelevinsky, and perhaps
others. This identity appears in [6, Exam. I.5.27(a), p. 93℄[12, Exer. 7.27(e)℄
and was given a bijetive proof by Sagan and Stanley [11, Cor. 6.4℄:
Proof.
X
s= =
Q
i (1
xi ) 1
Q
X
i<j (1
xi xj )
s= :
Apply the homomorphism ex that takes the power sum symmetri funtion pn
to Æ n u, where u is an indeterminate. This homomorphism is the exponential
speialization disussed in [12, pp. 304{305℄. Two basi properties of ex are
the following:
X
un
ex(f ) =
[x x xn ℄f
n!
n
1 2
1
Q
ex Q
= eu 2 u ;
i (1 xi ) i<j (1 xi xj )
1
1
2
0
+
where [x x
1
2
xn℄f denotes the oeÆient of x x xn in f . Sine
[x x xn ℄s= = f = ; when j=j = n;
1
1
we obtain
2
2
X un X
u2
k
X
uj X
f = :
n
!
j
!
j
n
`n k
`k j
Taking the oeÆient of un =n! on both sides yields (9). 2
0
f = = eu
+
1
2
=0
+
7
(10)
2.3 Corollary.
XX
n 0 Proof.
We have
un
N (n + jj; )s =
n!
X
!
s e p
u+ 12 u2 :
( 1 +1)
Multiply (10) by s and sum on to get
XX
n 0 N (n + jj; )s
un
= eu
n!
1
2
= eu
1
2
+
+
=
=
=
u2
u2
1 2
eu+ 2 u
X uj X
f = s
j
!
j
j=j j
X uj X
h
pj ; s= is
j ! j=j j
j
0
=
0
=
1
X uj X
j!
hpj s ; sis
1
j 0
j=j=j
j
Xu X
1 2
eu+ 2 u
pj1 s
j
!
j 0
!
X
1 2
s e(p1 +1)u+ 2 u :
2
The ase when onsists of a single row (or olumn) is partiularly
simple, sine then eah (~
; 1k j ) = 1 in (3). We will then write N (n; k) as
short for N (n; (k)). The oeÆient ej () beomes simply ej (k) = qj =(k j )!,
where j !qj is the number of permutations w 2 Sn with no yles of length
one or two. By standard enumerative reasoning (see e.g. [12, Exam. 5.2.10℄)
we have
1 2
X
j 0
qj xj =
e
x
1
2
x
x
:
(11)
From this and Theorems 2.1 and 2.2 it is easy to dedue the following results,
whih we simply state without proof.
2.4 Corollary.
(a) We have
N (n + k ; k ) =
k X
n
j =0
j
tn j =
8
k
X
j =0
(k
qj
j )!
tn
k j;
+
where qj is given by (11).
(b) Dene polynomials An (x) by A (x) = 1 and
0
An (x) = A0n (x) + (x + 1)An (x); n 0:
+1
Then
X
k0
N (n + k; k)xk =
() Let
e
1
2
u2 +2u
=
X
Then N (n + k; k) = bn if n k.
n0
bn
An (x)
:
1 x
un
:
n!
The stability property of Corollary 2.4() is easy to see by diret ombinatorial reasoning. If n k, then a skew SYT of shape =, where ` n+k and
` k, onsists of a rst row ontaining some j -element subset of 1; 2; : : : ; n,
together with some disjoint SYT U on the remaining n j letters. There are
tn j possibilities for U , so
N (n + k ; k ) =
n X
n
j =0
j
tn j ;
whih is equivalent to Corollary 2.4().
3
Asymptotis of
f =.
P
Rather than onsidering the sum `n f = , we ould investigate instead the
individual terms f = . The analogue of Theorem 2.1 is the following.
3.1 Theorem.
have
Let ` k and n k. Then for any partition ` n we
f = =
X
`k
z (; 1n k ) ( ):
1
9
(12)
The proof
P parallels that of Theorem 2.1. Instead of the power
sum expansion of `n s , we need the expansion of s (where ` n), given
by [6, p. 114℄[12, Cor. 7.17.5℄
Proof.
s =
X
`n
z ()p :
1
We therefore have
f = =
=
=
=
=
n k
p1 ; s=
*
+
X
n
k
1
s p1 ; z ()p
`k
*
+
n k X 1 s ; n k
z ()p
p1 `n *
+
X
s ; z(;11n k ) (; 1n k )(n k + m1 ( ))n p
`k
X
z(;11n k ) (n k + m1 ( ))n (; 1n k ) ( ):
`k
But
z ; n k (n k + m ( ))n = z ;
1
(
and the proof follows.
1
2
1
1
)
Theorem 3.1 an also be proved by inverting the formula given in [12,
Exer. 7.62℄.
We would like to regard equation (12) as an asymptoti formula for f =
when is xed and is \large." For this we need an asymptoti formula for
(; 1n k ) when is xed. Suh a formula will depend on the way in whih
the partitions inrease. The rst ondition onsidered here is the following.
Let ; ; : : : be a sequene of partitions suh that n ` n, and suh that
the diagrams of the n 's, resaled by a fator n = (so that they all have
area one) onverge uniformly to some limit ! . (See [2℄ for a more preise
statement.) We will denote this onvergene by n ! ! . The following
result is due to Biane [2℄, building on work of Vershik and Kerov.
1
2
1 2
10
Suppose that n ! ! . Then for i 2 there exist
onstants (dened expliitly in [2℄) Ci (! ), with C (! ) = 1, suh that for any
xed partition ` k of length `( ) we have
3.2 Theorem.
2
0
(; 1n k ) = f n
n
`( )
Y
1
k `( )) (1 + O (1=n)) ;
Ci (! )A n
1
(
2
+1
i=0
as n ! 1.
Let = z (; 1n k ) ( ). It follows
from Theorem 3.2 that
k2 =
=
=
O kn
, while = O k n
and = O k 2 n
for `( ) k 2. Hene if n ! ! then
1
(21
1 2
1
(1 )
f
n
=
(1 )
=
=
z
1
(1k )
n
f
(21
(1n) (1k ) + z
n
1
)
1 2
2)
(21n ) (21k ) (1 + O(1=n))
n
(21k
)
2
2
1 1
1
f +
C (! ) (21k ) p + O(1=n) :
k!
2(k 2)!
n
2
3
(13)
Let us note that by [6, p. 118℄[12, Exer. 7.51℄ the integer (21k ) appearing
in (13) has the expliit value
2
(21k
2
)=
f
P i 2
P 0 i
2
:
k
2
The leading term of the right-hand side of (13) is independent of ! , and
n
n
in fat it follows from [2℄ that f = k f f holds under thepweaker
hypothesispthat there exists a onstant A > 0 for whih n < A n and
`(n ) < A n for all n 1.
1
!
1
11
Given > 0, let
p
p
Par (n) = f ` n : (2 ) n < < (2 + ) n
p
p
and (2 ) n < `() < (2 + ) ng:
1
It is a onsequene of the work of Logan and Shepp [5℄ or Vershik and Kerov
[13℄ (see e.g. [1℄ for muh stronger results) that for any > 0,
X
2Par (n)
f tn ; n ! 1:
P
Thus not only is the sum N (n; ) = `n f = asymptoti to f tn =k! as
n ! 1 (as follows from (7)), but the terms f = ontributing to \most" of
the sum are \lose" to f f =k!.
Another way of letting beome large was onsidered by Vershik and
Kerov in [14℄ and in many subsequent papers (after rst being introdued
by Thoma). Let ; ; : : : be a sequene of partitions suh that n ` n and
suh that for all i > 0, there exist real numbers ai 0 and bi 0 satisfying
P
i (ai + bi ) = 1 and
1
2
n
lim i = ai
n!1 n
(n )0i
lim
= bi ;
n!1 n
where (n )0i denotes the ith part of the onjugate partition to n (i.e., the
length of the ith olumn of the diagram of n ). We denote this situation
by n ! (a; b), where a = (a ; a ; : : :) and b = (b ; b ; : : :). For instane, if
n = (n; n) and n = (n; n 1), then n ! ((1=2; 1=2; 0; : : :); (0; 0; : : :)).
TVK
1
2
2
2
1
2
TVK
1
The following result is immediate from [14℄.
3.3 Theorem.
(; 1n k ) =
n
Let n
`( )
Y
n
f
j =1
! (a; b). Then for any xed partition ` k,
TVK
X i j
i
+(
12
X 1)j 1
i j
i
!
(1 + O(1=n)) :
It follows that from Theorems 3.1 and 3.3 we have for xed asymptoti formula
2
f
n
=
n
= f 4
X
`k
z
1
`( )
Y
X i j
i
j =1
` k the
!3
X 1)j 1
i j 5 (1 + O(1=n)):
i
+(
(14)
Now let s (x = y ) denote the super-Shur funtion indexed by ` n in the
variables x = (x ; x ; : : :) and y = (y ; y ; : : :) [6, Exam. I.23{I.24℄, dened by
s (x = y ) = !y s (x; y ) (where !y denotes the standard involution ! ating
on the y -variables only). (Note that our s (x = y ) orresponds to s ( y = x)
in [6℄.) It follows that the expansion of s (x = y ) in terms of power sums is
given by
X
s (x = y ) =
z ( ) (p (x) p (y )) :
`n
Hene from equation (14) we obtain the following result.
1
2
1
2
1
3.4 Theorem.
Let n
f
n
=
! (a; b). Then for a xed partition we have
TVK
= f s (a = b)(1 + O(1=n)):
n
An expliit statement of Theorem 3.4 does not seem to have been published before. However, it was known by Vershik and Kerov and appears in
the unpublished dotoral thesis of Kerov. It is also a simple onsequene of
Okounkov's formula [9, Thm. 8.1℄ for f = in terms of shifted Shur funtions.
The asymptotis of shifted Shur funtions is arried out (in slightly greater
generality) in [3, Thm. 8.1 and Cor. 8.1℄. A speial ase of Theorem 3.4
appears in [10, Thm. 1.3℄.
Theorem 3.4 an be made more expliit in ertain ases for whih the
super-Shur funtion s (a = b) an be expliitly evaluated. In partiular,
suppose that onsists of an i j retangle with a shape = ( ; : : : ; i )
attahed at the right and the onjugate 0 of a shape = ( ; : : : ; j ) attahed
at the bottom. Thus
= ( + j; : : : ; i + j; 0 ; 0 ; : : :):
1
1
1
1
13
2
Then (e.g., [4, pp. 115{118℄[6, (4) on p. 59℄)
s (a ; : : : ; ai =
b ; : : : ; bj ) = s (a ; : : : ; ai )s (b ; : : : ; bj )
1
1
1
Y
1
i;j
(ai + bj ):
In ertain ases we an expliitly evaluate s (a ; : : : ; ai ) or s (b ; : : : ; bj ), e.g.,
when a = = ai or b = = bj . See [12, Thm. 7.21.2 and Exer. 7.32℄.
Note also that when = = ; (so = (j i )) we have simply
1
1
1
1
s j i (a ; : : : ; ai =
(
)
1
b ; : : : ; bj ) =
Y
1
i;j
(ai + bj ):
Aknowledgement. I am grateful to Andrei Okounkov for providing
muh of the information about Theorem 3.4 mentioned in the paragraph
following the statement of this theorem.
Referenes
[1℄ J. Baik, P. Deift, and K. Johansson, On the distribution of the length of
the longest inreasing subsequene of random permutations, J. Amer.
Math. So. 12 (1999), 1119{1178.
[2℄ P. Biane, Representations of symmetri groups and free probability, Advanes in Math. 138 (1998), 126{181.
[3℄ S. Kerov, A. Okounkov, and G. Olshanski, The boundary of Young
graph with Jak edge multipliities, Internat. Math. Res. Noties 1998,
173{199; q-alg/9703037.
[4℄ D. E. Littlewood, The Theory of Group Charaters and Matrix Representations of Groups, seond ed., Oxford University Press, Oxford,
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