Vector invariants of symmetric groups in prime
characteristic*
S. A. STEPANOV
Abstract - Let R be a commutative ring with the unit element 1 and Sn be the symmetric group of
degree n 2::: 1. Let A~ denote the subalgebra of invariants of the polynomial algebra
with respect to Sn, The classical result of H. Weyl implies that if every non-zero integer is invertible in R, then the algebra A~n is generated by the polarized elementary symmetric polynomials of degree at most n, no matter how large m is. As it was recently shown by D. Richman, this result remains true under the condition that
ISnl
=
n! is invertible in R. On the other hand, if Risa field of prime characteristic p ~ n, D. Richman proved that every system of R-algebra generators of A~n contains a generator whose degree is no less than max { n, ( m+
p - n) / (p - 1)}. The last result implies that the above Weyl bound on degrees of generators no longer holds when the characteristic p of R dividesISnl, In general, it is proved that, for an arbitrary commutative ring R, the algebra A~n is generated by the invariants of degree at most max{n,mn(n - 1)/2}. The purpose of this paper is to give a simple arithmetical proof of the first result of D. Richman and to sharpen his second result, again with the use of new arithmetical arguments. Independently, a similar refinement of Richman's lower bound was given by G. Kemper on the basis of completely different considerations. A recent result of P. Fleischmann shows that the lower bound obtained in the paper is sharp if m
> 1
and n is a prime power, n = pa.1.
INTRODUCTION
Let m, n be positive integers, R be a commutative ring with the unit element 1, and Amn
=
R[x11 , ... ,xml; ... ;xln• .. , ,Xmn]be the algebra of polynomials in mn variables x;1 over R. The symmetric group Sn operates on the algebraAmn as a group of R-automorphisms by the rule
a(x .. )=x. (')' IJ 1,C1 J
Denote by A~n the subalgebra of invariants of the algebra Amn with respect to the group Sn and define the polarized elementary symmetric polynomials u,l' .. .,rm E A~n in n vector variables
(xu, ... ,xml ), ... , (x1n, ... ,Xmn)
---
'U DC 519.4. Originally published in Diskretnaya Matematika (2000) 12, No. 4 (in Russian). Received August 18, 2000. Translated by the author.456 S. A. Stepanov
by means of the formal identity
n
Il
(1+
x,f I+ ... +
xmjZm)=
1+
L
Urp . ..,rmz{I ... Zm rm.j=I l:<,;r1+ ... +rm:<,;n
The elements of A~n are usually called the vector invariants of Sn. If R is Noetherian, then it follows from the Hilbert-Noether finiteness theorem (see [6, 9]) that
A!':.
is a finitely gener-ated commutative R-algebra and Arnn is finitely genergener-ated as a module over A~n. Moreover, if every non-zero integer is invertible in R, then the invariants Ur1 , ... ,rm form a completesys-tem of generators of A~':. over R (see [l], p. 9; [17], p:37). In other words, every element u of the algebra A~n may be written as a polynomial in
1::;
r1+ ... +rm::; n,
with coefficients in R. The above system of generating invariants contains (m!n) - 1 ele-ments connected with each o,ther by different algebraic relations (see [4, p. 68] and [14]). This result was recently generalised by D. Richman [11] as follows.
Theorem 1. Assume that
ISnl
=
n! is invertible in R. Then A~n is generated as an R-algebra by the polarized elementary symmetric polynomials1::;
r1+ ... +rm::; n,
of degree at most n.
In particular, if R is a field of prime characteristic p
>
n, then n ! is invertible in R, and we arrive at the following result.Corollary 1. Let R be a field. If char R = 0 or char R
=
p>
n, then A~':. is generated as an R-algebra by the polarized elementary symmetric polynomialsIn this paper, we give a simple arithmetical proof of Theorem 1 based on polarisation of classical Waring's formulas [16] and closely related to the Wey! original proof [17] in the case where R contains the field of rational numbers
Q.
The result of Theorem 1 can be easily extended as follows. Let A= R[x1 , ...
,xm]
be afi-nitely generated commutative R-algebra, G be a finite group of R-algebra automorphisms of
A, and AG be the subalgebra of invariants of G. If z1, ... , Zm are commuting indeterminates, then we set
F(z 1, ... ,zm)
=
I]
(I+
a(x1)z1+
a(x2)z2 + ... + a(xm)Zm), crEGIf every non-zero integer is invertible in R, then it follows from the Noether theorem that
AG is generated as an R-algebra by the coefficients of F(z 1, ... ,zm)- Theorem 1 and the standard arguments based on the use of the Reynolds operator and the Noether map (see [8]; [12], p. 63; [17], p. 275) lead to the following theorem.
Theorem 2.
If
I GI
! is invertible in R, then AG is generated as an R-algebra by the coef-ficients ofF(z
1, •••,zm),
In other words, AG is generated over R by invariant polynomialsin x1, ••• ,xm of degree at most
IGI.
The results of Theorems 1 and 3 provide us with an efficient algorithm to compute a complete system of generating polynomial invariants under the condition that
IGI
! is invert-ible in R. There is another constructive proof of Theorem 3 based on different arguments also ascending to Noether ([12], p. 29). The upper bound on the degrees of a set of gener-ating polynomials for the algebra of invariants given by Theorem 3 is known as Noether's bound (see also [ 12], p. 28, and [ 13]).If
I
SnI
=
n ! is invertible in R, then the upper bound on degrees of R-algebra generators of A~n' stated in Theorem 1, is the best possible. In the case where R is an arbitrary commutat-ive ring, it is proved in [3] that the Weyl algebra A~"n is generated over R by polynomials of degree at most max{n,mn(n - 1)/2}. A similar result was also obtained by Richman (see [11], Prop. 7). On the other hand, this paper gives the following lower bound.Theorem 3. Let
a
be a positive integer, Sn be the symmetric group of degree n ~ 2, and R be a field of prime characteristic p.If
pa divides n, then every system of R-algebra generators of A~n contains a generator whose degree is no less than max{n,m(pa -1) }.
This result sharpens the above mentioned Richman's lower bound in the case where p is a divisor of n, and shows that Noether's upper bound is false if n is not invertible in R. As it was recently proved by P. Fleischmann [5], the lower bound in Theorem 3 is exact if n
=
pa and m>
1.The result of Theorem 3 can be easily extended as follows. Let r be a positive integer that does not exceed n. In that case, the group S, is a subgroup of Sn and therefore A~n ~ A~n. This observation and Theorem 3 applying to S, yield the following result.
Corollary 2. Let A and r
s;
n be positive integers, Sn be the symmetric group of de-gree n ~ 2, and R be a field of prime characteristic p. If/' divides r, then in any sys-tem of R-algebra generators of A~n there exists a generator whose degree is no less than max{n,m(p,,. -l)}.
Corollary 3. Let Sn be the symmetric group of degree n ~ 2 and R be a commutative ring with the unit element I. If p is a prime divisor of n ! which is not invertible in R, then every system of R-algebra generators of A~"n contains a generator whose degree is no less
than max{n,m(p-
l)}.
If R
=
Z is the ring of integers, then we can use the well-known results on the distribution of primes in short intervals to get an universal lower bound in terms of m and n.Corollary 4. Let R
=
Z be the ring of integers and Sn be the symmetric group of degree n ~ 2. Then every system of R-algebra generators of A~n contains a generator whose degree is no less than max{n,cnm(n - l)}, where Cn = 1/2for every n ~ 2; moreover, Cn = 5/6 for every n ~ 25, and Cn -+ 1 as n -+ oo. In particular, if n is a prime number, then every458 S. A. Stepanov
system of R-algebra generators of A~'h co~tains a generator whose degree is no less than max{n,m(n-1)}.
A similar result holds in the case where R is the ring ZK of integers of a number field K.
2. GENERATING INVARIANTS OF THE SYMMETRIC GROUP Sn
Let Sn be the symmetric group of degree n ~ I that operates on the R-algebra Amn =R[x11 , ••• ,xm1; ... ;x1n,··· ,Xmn]
as a group of R-automorphisms, A~n be the subalgebra of invariants of the algebra Amn, and
u,1,···,'m'
1:::;
r1 + ... +rm:::; n, be the polarized elementary symmetric polynomials inA~'h-Let Vs1 , ... ,sm be an invariant polynomial in A~n of the form
If m
=
I, then according to the well-known Waring formula (see [16], p.13, and [2], p. 99)n
Vs=Lxj=
L
c(k1, ... ,kn)u~I ... u~", (1)j=I k 1 +2"2+ ... +nkn=s
where ck1 , ... .kn are integers of the form
(k k ) _ (-1)"2+2k3+ ... +(n-l)kn (k1 +···+kn -1)!
C 1, ... ' n - k I k I . I " .. n·S
The following result can be considered as a generalisation of the Waring formula to the case where m
>
1 (see also [14], [15]).Proposition 1. Lets 1, ••• , Sm be non-negative integers,
n
_ ~ SI x''m Vs!, ... ,sm - L.J Xlj ... mj
}=I
be the polynomial in A~n of degrees= s1 + ... +Sm, and u,., ... ,rm, 1 :::; r1 + ... +rm:::; n,
For non-negative integers k1, ... ,kn and k1v, ... ,kmv such that let k1
+
2kz+ ". +
nkn=
s' klv+ ...
+kmv=
Vkv, 1 ~ V ~ n,~
kv!nkv
Ivµ wklv•···,kmv=
£..J l ' l ' urlµ'"'>rmµ' Am,n vi·'·· Vkv' µ=Iwhere the sum is over the set Am,n of all non-negative integers r1µ,.,, , rmµ and lvi, ... , lvkv such that r1µ+ ... +rmµ=V for 1 ~ i $ m, 1 ~ µ ~ kv, 1 $ V ~ n. Then s,, ... sm,
r
r
nn V51 , ... ,sm=
c(k1 , ... ,kn) Wk k , s! Iv>"'' mv k1 +2ki+ ... +nkn=s Bm,n V=Iwhere the inner sum is over the set Bm,n of all non-negative integers kil, ... , kin satisfying the relations
kil + ... +kin =si, k1v
+ ...
+kmv=
Vkvfor 1 ~ i $ m, 1 ~ v $ n. Proof In (1) we set
Since
we have
On the other hand, V
. L . TI
(x1J/1+ .. · +
xmJ/m) =L
Ur1 , ... ,rmZ~1 .. • z;;,:,460
and hence, in view of (1),
As a result we find that
Now since where k I kv W
=
~
V •n
U~vµ r ' k1v···· ,kmv £.J [ ! ... / ! Iµ'"'' mµ Cm,v vi Vkv µ=Iand the summation is over the set Cm,v of all non-negative integers r 1µ,··· ,rmµ and lvl ' ... 'lvkv such that we find that 7ii lvl
+ · · · +
7is)vkv=
kiv• r 1µ+ ... +rmµ =V, lvl + ... +lvk =kv, . V 1 $ i $m, 1 $ µ $ kv,where the summation in the last sum is over the set Bm n of all non-negative integers kil, ... ,kiv• i
=
1, ... ,m such that 'kil
+ ...
+k;v=
S;, k1v+ ... +kmv=
Vkv,for 1 :S i :S m, 1 :S v :S n. Thus we arrive at the relation
n s1! ... sm! ~ (k k) ~
nn
L~1···rn0=
s! L.J C i , · · · , n L.J wklv•···,kmv'j=I k1 +2"2+ ... +nkn=s Bm,n V=I
which proves the proposition. Ifs=s1 + ... +sm :Sn+l,then
sl! ... sm! ~ ~
nn
Vsp···,sm
=
s! L.J c(k1,··· ,kn) L.J Wklv•···,kmvk1 +2"2+ ... +nkn=s Bm,n V=I
is a polynomial in ur1 , ... ,rm, 1 :S r 1 + ... +rm :S n, with rational coefficients whose denom-inators are not divisible by any prime p ~ n + 1. As a consequence of this observations we get the following result.
Corollary 5.
lf
n! is invertible in Rands= s1 + ... +Sm :Sn+ 1, then nVsp···,sm
=
LX~
1 .••-X:,.j
j=I J
is a polynomial over R in Ur1 , ... ,rm, 1 :S r1 + ... +rm :S n, of degree at most n.
Now we show that if n! is invertible in R, then any vector invariant in A~':, can be rep-resented as a polynomial over R in the invariants Vsl' ... ,sm.
Proposition 2. Let f be a monomial in Arnn and
v=
r
u.uE{ cr(J)JcrESn}
If
n! is invertible in R, then vis a polynomial over R in the invariantsn
Vs!''" ,sm
=
L ~
1. · ·-X:,.j,
j=I Jwhere s1, ••• ,sm are non-negative integers satisfying the condition 0 :S s1 + ... +sm :S deg/.
Proof We represent f in the form f
=
f1 .•• fn, where each fj is a monomial inR[x
1j, ...,xml
We setd(f)
=
mii.x (degfj) 1:::;1:::;nand prove the assertion by induction on S(f)
=
deg/ -d(f). Suppose at first that S(f)=
0. Then f=
fj=
~J ...
x'aj'
where j E { 1, 2, ... , n} and s I> ... , Sm are non-negative integers with the conditions 1 + ... + Sm=
deg/, and462 S. A. Stepanov
Suppose now that
8 (!)
>
0 and let j E { 1, 2, ... , n} be such thatd(!)
=
degfJ
< degf.
we define vJ and vJ, setting
The induction hypothesis implies that vJ and vJ are polynomials in
For every TE Sn, we define Ur: to be the set of all pairs
(u, u')
such thatu
E{er(!)
I
er
E Sn},u'
E{er(f/JJ)
Jer
E Sn},uu'
=
r(f)
and note that the map
(u,u')-+
(r(u), r(u'))is a bijection. Thus
IU,J
=
JUidl for all TE Sn. Note also thatd(uu')
2
d(f)
for allu
E{ er(!) I
er
E Sn} andu'
E{er(!/ J)
I
er
E Sn}, where the equality is attained if and only ifuu'
E {er(!)
Jer
E Sn}. Therefore,vJvJ
=
IUidl
L
u+L
L
u.uE{ a(J)laESn} J': degf'=degf,d(J')>d(J) uE{ a(J')laESn} By the induction hypothesis, the invariant
v J vJ -
I
uidlI,
u=
I,
I,
uuE{ a(J)laESn} !': degf'=degf,d(J')>d(J) uE{ a(J')laESn}
is a polynomial over R in v,p .. ·,'m' 0
:S
s1+ ...
+sm:S degf.
The cardinality of Uid doesnot exceed the cardinality of {
er(!)
Ier
E Sn} and the last cardinality does not exceed the cardinality of { x~1
.. .
x>,;;J I 1:S
j:S
n}, therefore 1:S
I Uid I:S
n. Since n ! is invertible in R, we conclude thatIi
u uE{ a(J)laESn}is a polynomial over R in v,1 , ... ,sm, 0
:S
er1+ ... +
erm
:S
degf. This completes the proof.Let Sn be the symmetric group of degree n
2
2 andbe a homogeneous polynomial in A~n of degrees 1
+ ... +
Sm2
1. Let R be a field of primecharacteristic p dividing n. The following result shows that Wey! 's bound fails to be correct over R form
>
n2
2.Proposition 3. Let R be afield of prime characteristic p and Sn be the symmetric group ofdegreen ~ 2. /fpdivides n, andm
> n, then the elementv
11... 11 EA~ cannot be expressed as a polynomial over R in the invariants1
:S
r1 + ... +rm :Sn.Proof Assume, for a contradiction, that v I I... I I is expressible as a polynomial over R in
u,J> .. .,rm• 1
:S
r1 + ... +rm '.Sn, and write v11 ... l I in the formn Sy
V11...11
=
L
asp···,snL
n n
Urlav•···•'mavs1 +2s2+ ... +nsn=m Rm(s1 , ... ,sn) V=I O"v=I
with some asp···,sn ER, where Rm(s1, ... ,sn) is the set of all non-negative integers rd, i
=
1, ... ,m, j = 1, ... ,sv, v = 1, ... ,n such that'iav + ... +rmav
=
V, 1:S
O"v'.S
Sv, 1:S
V '.Sn,';a1
+ ... +
';a.=
1, 1:S
i'.S
m.Without loss of generality we may assume that if k
:S
n is the smallest positive integer such that sk ~ 1, thenr lak -- {0 1 if ak
=
1, if2 :Sak:S
sk.Differentiating the above equality with respect to x11 and taking into account that OUTp···•'m
{o
ox
=
u(l,o, ... ,o)11 0,T2, ... ,Tm
if '1
=
0, if '1=
1,where u(l,o, ... ,o) is the corresponding elementary symmetric polynomial of vectors
O,r2,···,'m
(x2i' ... ,xm)• 1
:S
j :Sn, we obtainn
x x
=
~ a ~ q,U)21 · · · ml L..i si,···,sn L..i s1, ... ,sn'
s1 +2s2+ ... +nsn=m j=I
(2)
where q,U)
S1 , ... ,Sn
We denote by w0,Tz,··· ,Tm the value of u0,Tz,··· ,'m at the point
(x11,··· ,xml; ... ;xln•··· ,Xmn)
= (1, ... , l; ... ;l. .. ,
1).Since m
>
n ~ 2, each binary sequence (0, r 2 , ••• , rm) encountered in the last equalitycon-tains l non-zero elements for some 1 ~ l ~ n. In that case <.Oo. , =n(n-1) ... (n-l+l),
464 S. A. Stepanov
and setting
x, I
= ... =
xml= ... = x,n = ... =
Xmn=
1 in (2), we arrive at the relation1
=
nL
bsi,···,sn, s1 +2s2+ ... +nsn=mwhich is impossible in R for any prime p dividing n. This completes the proof.
3. PROOF OF THEOREM 1
Let Sn be the symmetric group of degree n. Suppose that f is a monomial in Arnn and
w E A~"n is a polynomial invariant of Sn. Since cr(w)
=
w for any er E Sn, the polynomials wand cr(w) have equal coefficients. This shows that every polynomial invariant of Sn is an R-linear combination of the invariantsv=
L
u,uE{ cr(f)lcrESn} where f varies over the monomials which appear in w.
Let (ipi2 , •.. ,iµ) be a sequence of elements i1,i2 , ••• ,iµ E {1,2, ... ,n}. At first we prove that every invariant wµ of the form
n
Wµ
=
Lxi1,i"" .xiµ,ij=I
is a polynomial over R in polarized elementary symmetric polynomials
Ifµ ~ n
+
1, then the assertion of Theorem 1 follows from Proposition 1. Assume now thatµ
>
n
+
1 and proceed the proof by inductiononµ.
Let{ X· . - ls,1 X·
·=
ls,1 X· ·X· · X· · 1n+1'1 1n+i,1 ·' · 1µ+1'1 ifs ~n, ifs= n+ 1 for j= 1,2, ...
,n,
and writeLet
n
Wµ+I
=
LXil>j · .. iin,/in+l'j·j=I
Amn
=
R[i11 , ... ,iml; ... ;iln• ... ,Xmn]and let A~"n be the subalgebra of invariants of Arnn. It follows from Corollary 5 that Wµ+I is a polynomial over R in the polarized elementary symmetric polynomials
Since every such polynomial u,1>···,'m has the form
U,1' ... ,rm ==
L
U,uE{-r(l)l-rEG}
for some monomial
J
EAmn
of degree at most n, by Proposition I it can be written as a polynomial over R in the invariantsn
-
-
~.i51
~mVs1>··· ,sm - L.J lj ... mj
}=I
of degree at most n. Moreover, each iis1 , ... ,sm has the form
n
v
=
~ ;:/1 .. ./1
s1>··· ,sm L.J lj mj'
}=I
where I S:: t 1
+ ... +
tm S::µ.
The induction hypothesis implies that every invariant Vs 1, ••• ,m isa polynomial over R in
therefore wµ+I is also a polynomial over R in the polarized elementary symmetric polyno-mials Ur1,--..rm• 1 S:: '1
+ ... +rm
S:: n.To complete the proof, we note now that every element v E A~n can be written, in view of Proposition 2, as a polynomial over R in the invariants
wµ-4. PROOF OF THEOREM 3
The arguments which we shall use are the same as in the proof of Proposition 3. Let Sn be the symmetric group of degree n ~ 2, and suppose that p is a prime divisor of n. Let R be a field of characteristic p and A~n be the algebra of vector invariants over R with respect to Sn. Let
denote a polynomial in A~~ of the form
Recall that every vector invariant v EA~~ is an R-linear combination of the invariants
W=
L
u,uE{ a(f)laESn} where f varies over the monomials in Amn which appear in v.
To prove Theorem 3, it is sufficient to show that if pa
In,
then every system of R-algebra generators of A~n contains at least one generator v of degree m(pa - I). The crucial point is that the invariant v pa-I, ... ,Pa_ 1 of degree m(pa - 1) cannot be presented as a polynomial466 S. A. Stepanov
We denote by W~~ the set of R-algebra generators of A~"n of the form
W=
L
u,uE{ a(f)laESn}
where f is a monomial in Amn of degree less than m(pa - 1). Since every vector invariant v E A~n whose degree is less than m(pa - 1) can be written as an R-linear combination of elements w E W~~. it suffices to prove that the invariant vpa-l, ... ,pa_ 1 is not representable as a polynomial over R in the elements w E W~~- Let l denote the cardinality of W~~- We enumerate the elements of W~~ by the numbers 1, 2, ... , / and assume, for the contradiction, that the invariant vpa-l, ... ,pa_ 1 is a polynomial over R in w1, ... , w1, that is,
where Mis the set of all non-negative integers µ1, .•• , µ1 such that 0 $ µ1
+ ... +
µ1 $ m(pa - 1), µ1 degw1 + ... + µ1degw1=
m(pa - 1).(3)
Comparing the degrees of the monomials which appear in both sides of the last iden-tity (with respect to each of the variables x 11 , ... ,xml; ... ;x1n, ... ,Xmn), we find that µk E {O, 1, ... ,pa - 1 }, 1 $ k $ l, and µ1
+ ... +
µ1>
1; moreover, every invariantwk=
I
u,uE{ a(fk)laESn}
which appears in the right-hand side with a non-zero coefficient, is generated by a monomial fk E Amn of the form
where 1 $
pl
$ ... $ }. (k) $ n and}. (k) $ pa - 1 at least for one i, i=
1,2, ... ,m.'iii ,vi ,vi
We denote by wk the value of wk at the point
(x11 , ... ,xm1; ... ;x1n,··· ,Xmn)
=
(1, ... ,1; ... ;l, ... ,1) and observe that wk= lorb(fk)I, whereorb(fk)
= {
<J(fk) I <J E Sn} is the orbit of fk under Sn. IfSince }. (k) :::; pa - 1 for every k = 1,2, ... ,l and at least for one i, i = 1,2, ... ,m, the ,vi
exponent of pin the prime factorisation of every ISn(/k)I is less than the exponent of pin the prime factorisation of n !. This implies that lorb(fk) I is divisible by p, therfore rok = 0 in
R for all k = 1, 2, ... , l. Differentiating now identity (3) with respect to x11 , setting X11 = ... =Xml = ... =X1n = ... =Xmn = 1
in the resulting identity, and taking into account that every product
in the right-hand side of (3) involves at least two factors, say wk and wk'' with 1:::; k:::; k':::; l,
we arrive at the relation p - 1
=
0, which is impossible in R.Since the invariant vpa-i, ... ,pa_1 cannot be written as a polynomial over R in vector
invariants of smaller degree, every system of R-algebra generators of
A~"n
has to contain at least one generator of degreem(pa -1). Observing now that every system of R-algebra gen-erators ofA~"n
contains a generator of degree n (for example, the invariant w = x11 ... x1n),we find that it has to contain a generator v whose degree is no less than max { n, m(pa - 1)}. 5. PROOF OF COROLLARY 3
Let R be a commutative ring with the unit element 1, let p be a prime divisor of n ! , and m be a maximal ideal in R that contains p. Then F = R/m is a field of characteristic p. Consider the reduction homomorphism
<p: R[x11 , ••• ,xml; ... ;xnl, ... ,Xmn]
---+
F[x 11 , ... ,xml; ... ;xnl, ... ,Xmn],which leaves fixed all the variables xij, 1 :::;
i:::;
m, 1 :::; j:::; n. This homomorphism inducesa surjective homomorphism
1/f: R[x11 , ... ,xml; ... ;xnl, ... ,Xmn]5•
---+
F[x11 , ••• ,xml; ... ;x1n, ... ,Xmn]5•.Therefore 1/f maps every set of generators of R[x11 , ... ,xm1; ... ;x1n, ... ,Xmn]5• to a set of generators of F[x 11 , ... ,xm1; ... ;x1n, ... ,Xmn]5•. This fact and Corollary 2 imply that every system of R-algebra generators of R[x11 , ... ,xm1; ... ;x1n, ... ,Xmn]5• contains a generator of degree at least max{n,m(p-l)}. In particular, if n =pis a prime number, we conclude that it contains a generator whose degree is no less than max{n,m(n - 1)}.
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468 S. A. Stepanov
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