Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://www.mccme.ru/~smirnoff/papers/dubna14.pdf Äàòà èçìåíåíèÿ: Sat Jan 16 07:43:29 2016 Äàòà èíäåêñèðîâàíèÿ: Sun Apr 10 12:56:38 2016 Êîäèðîâêà:

« » ,

. .

,




C

.

.+ . +.

C

. . , . -- .: , .-- . ISBN - , , , ? , , , . , , , . , « » . .



.

+

.

: . . , . .

ISBN

--

-

-

© . ., © , .

.



. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... ... ... ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. q- . . . . . . . . . . . . . . . . q- . . . . . . q- . . . . . . . . . . . . . . . . q- . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

....... ....... ....... ....... ....... ....... . . . . . . . . . . . .. .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ---- . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . -- . . . . . . . . . . . . . . . . . . . . . . . -- . . « » . . . . . . . . . . . . . - . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.. .. .. .. .. .. ..

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .



. , , , . p (n) n -- , -- : . , - -- ! -- - . , , -- p (n)q n -- :


p ( n) q n =
n =0

(1 - q k )-1 .
k =1

XVIII . : , . , , . . . , , . . , 5 + 3 + 3 + 1 12 :

, p (n) -- , n .




: , , « » ( ) , , , (. . « »). , , .

, . XX . . , . , , , . -- (alternating sign matrix conjecture), - . , , . . ( -- «» ), - . . . . , , , ±1. , , , . . , XIII « » . -




. ; , , . . , ; . ; , . , , . . , , - . . , « ». , , . . . q - , q - -- . , , , , . . [ ], , , . ( ) , q -, .




( ) -- - . ---- . , ; , ( ) , , : (TSSCPP) . . . . . . . , , , . , . . . , . . . . . , «» . . , . esmirnov@hse.ru. MathNet, http://www.mathnet.ru.





§ . .
. : , , 2 + 3 3 + 2 -- . . . . . n = (1 , ..., k ), 1 2 ... ... k > 0 1 + ... + k = n . , -- (1 , ..., k , ...), i . , i «» . . (1 , ..., k ) , . , , i - i ( , ). . . , (7, 5, 5, 5, 2, 1) .
; , . , 45 , , y = | x |. « » y = f ( x ). , , , . . , ( ., , [ ]).


§ . .

(. . x + y = 0). , = (1 , ..., ) m ( ). , i , i . . . (6, 5, 4, 4, 4, 1, 1) . n p (n). , p (0) = 1. p (n) n (, n 5). .
n p (n )

. . n 5 , n. : - , p (n) n? , (, , ) . - : , . .

§ . .
. -- , [ ] [ ]. a0 , a1 , ..., an , ... -- . q : a0 + a1 q + a2 q 2 + ... + a n q n + ... ( )


.

. . . -- (), « ». , , . , : , , , , , . q . . . A(q ) = a0 + a1 q + a2 q 2 + ... -- , a0 = 0. , B(q ), A(q ), -- . . , A(q ) · B(q ) = 1. a0 = 0? , , - (, q), - (a0 , ..., an , ...). . . n -- n , ak = . , ak = 0 k > n. k
n n n2 nn + q+ q +...+ q = (1 + q)n . 0 1 2 n

. . . : an = 1 n 0. : 1 + q + q2 + ... + q n + ... = 1 - q . : 1 - q ( . ) 1 + q + q 2 + ...: , . . . , an = n + 1 1 + 2q + 3q 2 + ... + (n + 1)q n + ... =
1 (1 - q)2 1


§ . .

( ). , , . . . ( ). an an+2 = an+1 + an , a0 = = a1 = 1. -- F (q ) = 1 + q + 2q 2 + 3q 3 + 5q 4 + 8q 5 + 13q 6 ... an , F (q ) = 1 + q F (q ) + q 2 F (q ) (, ). F (q ) : F (q ) =
1 . 1 - q - q2

. . a . ,
q-b

. , ( ). . : n , m? pm (n), Pm (q ). , P1 (q ) = 1 + q + q 2 + ... = 1 - q : . , p2 (n), . . n . , n , , -- , n , , n . 1 + q2 + q4 + ... =
1 . 1 - q2 1


.

, P2 (q ) P1 (q )
1 . , 1 - q2

(1 + q + q 2 + ...)(1 + q 2 + q 4 + ...). , . q r · q 2s , q r , q 2s . r + 2s r s . , q n p2 (n). , . . . Pm (q ) n , m, Pm (q ) = pm (n)q n =
1 = (1 - q)(1 - q2 )...(1 - qm )
m

(1 - q k )-1 .

k =1

, , m. , m . . . (. ). P (q ) n : P (q ) = p ( n) q n =
1 = (1 - q)(1 - q2 )(1 - q3 )...


(1 - q k )-1 .
k =1

. . , . , . , , , , « ». (, k -) , ( k ) -- q k .


§ . .

§ . .
, . : ? pO (n) ( «odd» -- «»). , pO (7) = 5, : 7 = 5 + 1 + 1 = 3 + 3 + 1 = 3 + 1 + 1 + 1 + 1 = 1 + 1 + 1 + 1 + 1 + 1 + 1. , , . pD (n) ( «distinct» -- «»). : 7 = 6 + 1 = 5 + 2 = 4 + 3 = 4 + 2 + 1. ! , . . . n pD (n) = = pO (n). . , PO (q ) = pO (n)q n PD (q ) = pD (n)q n . , , , PO (q ) = pO (n)q n PO (q ) =
1 1 1 · ... · 1 - q 1 - q3 1 - q5

: PD (q ) = (1 + q )(1 + q 2)(1 + q 3 )... ( ). . PO (q ) (1 - q 2 )(1 - q 4 )...: · ... = 1-q · · ...= PO (q ) = 1 - q · 1 - q3 1 - q5 1 - q2 1 - q3 1 - q4 =
(1 - q2 )(1 - q4 )(1 - q6 )... = (1 + q )(1 + q 2)(1 + q 3 )... = PD (q ). (1 - q)(1 - q2 )(1 - q3 )... 1 1 1 1 1-q
2

1

1-q

4


.

: , , . - «» () ? , n ? . . - , , : 23 = 7 + 5 + 5 + 3 + 1 + 1 + 1. « » -- (. . ), , -- .

. . .

, . . . 23 : 23 = 10 + 6 + 4 + 2 + 1.

§ . .
, P (q ). :


P (q )

-1

=

(1 - q k ).
k =1


§ . .

. . ( q 7 ) . , : P (q )
-1

= 1 - q - q 2 + q 5 + q 7 - q 12 - q 15 + q 22 + q 26 - q 35 - q 40 + ...

. , , , ±1, . -, , , : , , . . : (q q 2 ), (q 5 q 7 ), (q 12 q 15 ) . . , 1, 5, 12, 22, 35, ... -- , , 1, 2, 3, ... (. . . ).

. . .

, m- . 2 , . . . ( ). , P (q ) = p (n)q n ,


m(3m - 1)

P (q )

-1

= 1+

(-1)
m =1

m

q

m(3m-1) 2

+q

m(3m+1) 2

.

. -


.

, , [ ] [ ]. , . P (q ) P (q )-1 , .


(

p (k )q k ) 1 +

(-1)
m =1

m

q

m(3m-1) 2

+q

m(3m+1) 2

= 1.

, q n n > 0 P (q ) P (q )-1 . , an q n bm q m -- . q k ak b0 + a
k -1 b1

+ ... + a0 bk =

a

k -i bi

.

, q n , p (n):


p ( n) +

(-1)
m =1

m

p n-

m(3m - 1) m(3m + 1) + p n- 2 2

=0

( , p (k ) = 0 k < 0). , , , ,


p ( n) =

(-1)
m =1

m +1

p n-

m(3m - 1) m(3m + 1) + p n- 2 2

.

, . 6 n 12 p (n) ( n 5 -- . . ). p (6) = p (5) + p (4) - p (1) = 7 + 5 - 1 = 11; p (7) = p (6) + p (5) - p (2) - p (0) = 11 + 7 - 2 - 1 = 15; p (8) = p (7) + p (6) - p (3) - p (1) = 15 + 11 - 3 - 1 = 22; p (9) = p (8) + p (7) - p (4) - p (2) = 22 + 15 - 5 - 2 = 30; p (10) = p (9) + p (8) - p (5) - p (3) = 30 + 22 - 7 - 3 = 42; p (11) = p (10) + p (9) - p (6) - p (4) = 42 + 30 - 11 - 5 = 56; p (12) = p (11) + p (10) - p (7) - p (5) + p (0) = 56 + 42 - 15 - 7 + 1 = 77.


§ . .


. . a) (. . ) n . ) . . ( ). , § . n . . . an -- n. , 1 +
q2 . 1 - 2q

. . n n, n; , , 6 = 1 + 2 + 3 + 6 = 12. (q ) -- n : (q ) = q + 3q 2 + 4q 3 + 7q 4 + 6q 5 + 12q 6 + 8q 7 + ... ) , ( q ) P ( q ) = q P ( q ), P (q ) -- . ) n . . P (q ): (ln P ( q )) = P ( q ) .
P (q )




q-
§ . .
: m , n? , m â n? , 2 â 2 . . , . , , . m + n , m , n . m m + n (, , n ), . .
m+n m+n = . m n

§ . . q-
|| . -- , -- . ,
m+n m

:
q

m+n m

=
q

q || .
m â n


§ . . q -

, q k k , m â n. , , q , q
m+n . m

q - . , , : «q -», . . q - . . . m = 1.
n+1 n

= 1 + q + q2 + ... + q n =
q

1 - qn+1 . 1-q

. . m = n = 2. 2 = 1 + q + 2q 2 + q 3 + q 4 q ( , , -- , , ). , : 1 + q + 2 q 2 + q 3 + q 4 = (1 + q + q 2 )(1 + q 2 ). q - . . .
m+n m

4


q

m+n . m

: q = 1, , . . . . .
m+n m+n = . m n

(. . ). . ( q - ).
m+n m+n-1 = +q m m-1
m

m+n-1 . m


. q -

. . ; . : , , , (m - 1) â n; , m â (n - 1), m.
m+n m = m+n-1 m+n-1 + m-1 m

. . . q - :
m+n m+n-1 = +q m m
n

m+n-1 . m-1

, q - . , :
1 1 1 1+q 1 1 1 1 + q + q2 1 + q + q2 1 1 + q + q2 + q3 1 + q + 2q2 + q3 + q4 1 + q + q2 + q3 1 1 ..............................................................................

, q - . = n m! · n ! , q -. . . n 0. q - n [n] = 1 + q + ... + q
n -1 m+ n

(m + n)!

= 1-q .

1-q

n

q - n [0]! = 1, [n]! = [n - 1]! · [n].

, [n] [n]! n n! .


§ . .

. . ,
m+n n

=

[m + n]! . [m]! · [n]!

. [m + n] = [m] + q m[n]. . , [k ] 1 - q . . .
m+n m 1-q
k

=

(1 - qm+n )(1 - qm+n-1 )...(1 - qn+1 ) . (1 - qm )(1 - qm-1 )...(1 - q)

q - , , : . ,
m+n m

q k

q mn-k . q - : k m â n, mn - k . q - : ( ) . , (a0 , ..., amn ) : mn , . . ai a j 0 i < j . 2 . . ' [ ]; . . [ ] . . [ ].

§ . . q-
, n


. q -

m. n ; ,
n

lim

m+n m

=

1 (1 - qm )(1 - qm-1 )...(1 - q) m+n , m

(,

. , n !). . , m. , n m (, lim
n

2n ) -- n

( . ).

§ . . q-
x k q -. (1 + x )n = k ; . . (q - ).
n k =0 n

(1 + xq )(1 + xq 2)...(1 + xq n ) =

n q k

k(k+1) 2

xk.

, q = 1 . . x , q : (1 + xq )(1 + xq 2)...(1 + xq n ) =
n k =0

ak (q ) x k .

ak (q ) -- k , n. ( , ) k , k - 1, ..., . q 1+2+...+k = q 2 . k
k(k+1)


§ . .

(, , ), n - k . q - ak (q ) = q
k(k+1) 2

n . . k

n . k

§ . .
q - , . , , . , :
m

(1 + x q i ) = = (1 + x q
-(m-1)

i =-(m-1)

)(1 + x q

-(m-2)

)...(1 + x )(1 + xq )...(1 + xq m ),

x -- . . q -. , : y = xq -m .
m

(1 + x q i ) =

2m

(1 + x q
i =1

i -m

2m

)=

(1 + y q i ).

i =-(m-1)

i =1

, q -.
2m

(1 + y q i ) =

2m k =0

i =1

2m q k

k(k+1) 2

y k.
-m

: y q (1 + x q
1-m

x ,

)(1 + x q

2-m

)...(1 + x )(1 + xq )...(1 + xq m ) =
2m

=
k =0

2m q k

k(k+1) 2

-mk k

x.

q (. . , q , ). ,


. q -

q ( m). q , q , . , 1 + xq - j , j 0, , 1 + xq - j = q - j ( x + q j ) = xq - j (1 + x -1 q j ) ( x , ). x mq
-(0+1+...+(m-1))

(1 + x

-1 m -1

q

)(1 + x

-1 m -2

q

)...(1 + x
2m k =0

-1

)â
k(k+1) 2

â (1 + xq )(1 + xq 2)...(1 + xq m ) = , x m q (1 + x
-1 m -1 -

2m q k

-mk k

x.

m(m-1) 2

,

q

)(1 + x

-1 m -2

q

)...(1 + x

-1

)(1 + xq )(1 + xq 2)...(1 + xq m ) =
2m

=
k =0

2m q k

k(k+1) 2

+

m(m-1) 2

-mk k -m

x

.

: , j = k - m. j -m m, q «» :
m

m ( m - 1) - mk . 2
-1 k -1 m

(1 + x q k )(1 + x

q

)=
j =- m

k =1

2m q m+ j

j ( j +1) 2

x j.

. m . P (q ) = k=1 (1 - q k )-1 (, , j , ). P (q ), . . ( ).
+

(1 + x q k )(1 + x
k =1

-1 k -1

q

)(1 - q k ) =
j =-

q

j ( j +1) 2

x j.


§ . .

. . : q 3 q .
+

(1 + x q 3 k )(1 + x
k =1

-1 3 k -3

q

)(1 - q 3 k ) =
j =-

q

3 j 2 +3 j 2

x j.

x = -q


-1

.
+ 3 k -2

(1 - q
k =1

3 k -1

)(1 - q

)(1 - q 3 k ) =

(-1) j q
j =-

3 j2+ j 2

.

:


(1 - q k ) = 1 +
k =1

(-1) j q
j =1

j (3 j -1) 2

+q

j (3 j +1) 2

.


. ( ). q = p m -- . , k - n- q
n . k

q - [ ]. . ( q - ). x y , y x = q xy . , ( x + y )n =
n k =0

nk xy k

n- k

.

, q - .


f (x) =

(1 + x q k )(1 + x
k =1

-1 k -1

q

).

, , , x y . , , x = 0 q y = 0 0 .
10 01


. q -

x :


f (x) =
n =-

an (q ) x n ,

an (q ) q . . . , f ( xq ) = x -1q -1 f ( x ). . . , n : ) a n ( q ) q n + 1 = a n + 1 ( q ) ; ) f ( x ) = a 0 ( q ) q n(n+1)/2 x n .
n

. . bm m {1, 2, 3, ...} {0, 1, 2, ...}. , a0 (q ) = bm q m .
m0

. . , , bm p (m), . . m. . . . . .





§ . .
«» . -- , , , , . . . n -- . -- i, j , i , j 1, n (. . i, j = n) i, j

i, j i+1, j i, j i, j +1 i , j . , -- , , , . . , . . . . . . .
3 2 2
. . .

2 1

1

1

µ1 µ2 ... µk . µm (i , j ), 11 - i, j m - 1.


.

,

µm m- «» ( ). . .

, , (a, b, c) (. . , a, b c ), . a, b, c , ( XX . ).

§ . .
: (a, b, 2). (µ1 , µ2 ) , µ1 µ2 (a â b). , a â b , A1 = (0, a) B1 = (b, 0) (, ). , -- A1 B1 , . : (1, 1). A2 = (1, a + 1) B2 = (b + 1, 1), (. . . ). , , Pnc ( A1 B1 , A2 B2 ) ( «nc» «non-crossing», . . «»).


§ . .

B

1

B

1

B

2

A

1

A

1

A

2

. . .

, A1 c B1 , A2 c B2 : P ( A1 B1 ) · P ( A2 B2 ), . .
a+b b
2

. , -

A1 B1 A2 B2 . , µ1 µ2 -- , A1 c B1 , -- A2 B2 . C -- . , (µ1 ) (µ2 ), : (µ1 ) µ1 A1 C µ2 C B2 . , A1 B2 . , (µ2 ) A2 B1 ; µ2 A2 C µ1 C B1 (. . . ). , , A1 B2 , A2 B1 , ; , , - µ1 : A1 B1 µ2 : A2 B2 . A1 B1 A2 B2 ,
B
1

B B
2

1

B

2

A

1

C A
2

A

1

C A
2

. . .


.

«» . P ( A1 B 2 ) · P ( A2 B 1 ) =
a+b a+b · . b+1 b-1

, . . . Pnc ( A1 B1 , A2 B2 ) = P ( A1 B1 ) P ( A2 B2 )- - P ( A1 B 2 ) P ( A2 B 1 ) = P ( A1 B 1 ) P ( A1 B 2 ) . P ( A2 B 1 ) P ( A2 B 2 )
2

. . , (a, b, 2),
a+b b - a+b b-1 a+b . b+1

, 2 â 2. , n .

§ . . ----
: (a, b, c). c . , , A = (0, a) B = (b, 0). . , : k - (k - 1, k - 1). k - Ak = (k - 1, a + k - 1) Bk = (b + k - 1, k - 1). w Sn -- . , c (, ) ( A1 , ..., Ac ) ( B1 , ..., Bc ) w , , Ak , Bw (k) . , . , w . -


§ . . ----

:
c

P ( Ak B
k =1 c

w (k )

)=
c

=
k =1

P ((k -1, a + k -1) (b+ w (k )-1, w (k )-1)) =
k =1

a+b . b + w (k ) - k

, : , , , . µ µi i , - . . , µi , ; µ j . µi , , µ j , . (µ). ; , , , , µ (µ), . , , : , . . , , , , . . . . . , : (2, 2) -- -
B
1

B B

1

B
2

2

B A
1

3

B A
1

3

A

2

A A
3

2

A

3

. . .


.

( ) -- . 1 3 2 , -- 2 3 1 . , w , :
c

123

123

P (( A1 , ..., Ac ) ( B
w S
n

w (1)

, ..., B

w (c)

)) =
w S
n

(sgn w )
k =1

P ( Ak B

w (k )

).

, , , . , , . , Pnc (( A1 , ..., Ac ) ( B1 , ..., Bc )). , , i - j - Ai B j . : Pnc (( A1 , ..., Ac ) ( B1 , ..., Bc )) = det P ( Ai B j )
c i , j =1

.

P ( Ai B j ), . . . P P (a, b, c) (a, b, c) : P P (a, b, c) = det
a+b b+ j -i
c

=
i , j =1

=

a+b b a+b b-1

. . .

a+b b+1 a+b b

... ... .. . ...

. . .

a+b b+c-1 a+b b+c

. . .

.

a+b b-c+1

a+b b-c+2

a+b b

, , . [ ] . . [ ],


§ . . ----

( , ). , -, - . . . [ ], [ ]. . [ ]. -- , . § . . . . . (a, b, c) P P (a, b, c) = det
a+b+i-1 b+ j -1 a+b b
c

=
i , j =1

a+b b+1 a+b+1 b+1

... ... .. . ...

a+b b+c-1 a+b+1 b+c+1

=

a+b+1 b

. . .

. . .

. . .

.

a+b+c-1 b

a+b+c-1 b+1

a+b+c-1 b+c-1

. . P P (a, b, c) Ak = (k - 1, a + k - 1) Bk = = (b + k - 1, k - 1), 1 k c. , , , B k , -- , B = (b + c - 1, k - 1) (. . . ). k
B
1

B B
2

1 2 3

B B B
3

A

A
1

1

A

A
2

2

A

A
3

3

. . .


.

, . P P (a, b, c) = det( P ( Ai Bj )) P ( Ai Bj ) =
a+b+c-i . b+c- j
c i , j =1

.

, . . . . . , . § . . , .

§ . .
. . ( ). 1x
1

x

1 x2 x ... x .. . . .. . .. . . . . . . n- 2 1 xn xn ... xn

2 1 2 2

...x

n -1 1 n -1 2

=
i> j

( x i - x j ).

1

. , . , , n. : , x1 . , , , . , . (k -) xk - x1 , .


§ . .

, , . . , xi x j . , xi - x j . xi - x j ( ). , n(n - 1)/2, . - n 2 (, x2 x3 ... xn -1 ) , , . , . . . n 0 . n- x x
n

= x ( x - 1)...( x - n + 1)

( n ; , , x 0 = 1). , nn = n!, x (n+m) = x n ( x - n)m . , :
n k

=

nk nk . = k! kk

, , . . . 1x
1

x

2 1

...x

n -1 1 -1

1 x 2 x 22 ... x 2 n .. . .. . .. . . . .. . . n 2 1 xn xn ... xn

=
i> j -1

( x i - x j ).

. . . . .


.

§ . .
§ . , (a, b, c)
a+b b a+b b+1 a+b+1 b+1

... ... .. . ...

a+b b+c-1 a+b+1 b+c+1

P P (a, b, c) =

a+b+1 b

. . .

. . .

. . .

=

a+b+c-1 b

a+b+c-1 b+1

a+b+c-1 b+c-1 a+b+i-1 b+ j -1
c

= det

.
i , j =1

. ( j -) 1/(b + j - 1)!. ,
c

P P (a, b, c) =
j =1

1 (b + j - 1)!

· det (a + b + i - 1)

b + j -1 c i , j =1

.

: (a + b + i - 1)
b + j -1

= (a + b + i - 1)b · (a + i - 1)
b

j -1

(a + b + i - 1) :
c

(i -) j -1 c i , j =1

P P (a, b, c) =
j =1

(a + b + j - 1)b (b + j - 1)!

· det (a + i - 1)

.

; 1! · 2! · ... · (c - 1)!. :
c

P P (a, b, c) =
j =1

(a + b + j - 1)

b

c j =1

( j - 1)! = (b + j - 1)!

c j =1

(a + b + j - 1)b . (b + j - 1)b

:
b c

P P (a, b, c) =
i =1 j =1

a+i+ j -1 . i+ j -1


§ . .

. : a, b c ( b c ). : ,
a+i+ j -1 a+i+ j -2 i+ j a+i+ j -1 = · ·...· . i+ j -1 a+i+ j -2 a+i+ j -3 i+ j -1

a , a, b c . , . . ( ). (a, b, c)
a b c

P P (a, b, c) =
i =1 j =1 k =1

i+ j +k-1 . i+ j +k-2

. ( ) (i , j , k ). ht = i + j + k - 2. , , ( ) 2 . . . . ( ). (a, b, c) P P (a, b, c) =
(a , b, c)

ht + 1 , ht

, (a, b, c). . . , () : a â b
a+b a

=

(a + 1)(a + 2)...(a + b) = 1·2·...·b

b j =1

a+ j = j

a

b

i =1 j =1

i+ j , i+ j -1

. . , (i , j ) i + j - 1.


.

, . , (, -, ).

§ . .
P Pq (a, b, c) , (a, b, c): P Pq (a, b, c) = q || .
(a , b, c)

, || -- , . . . : c = 1. -- a â b. , q - : P Pq (a, b, 1) =
a+b b

.
q

q- . . , k- , q k ( ). . P wt P , P . , P -- (0, a) (b, 0), q || , || -- , P (. . . ). q - :
a+b b a+b b

=
P : (0,a)(b,0)

wt P .

q = 1 , , .


§ . .

1 q q q q q
1

1 q q q q q
1

1 q q q q q
1

1 q q q q q
1

1 q q q q q
1

2

2

2

2

2

3

3

3

3

3

4

4

4

4

4

5

5

5

5

5

. . .

. . . . (0, 5) (5, 0), q 4 · q 4 · q = q 9 . . , . c. c , . q N , N -- . , § . . , k - (k - 1, k - 1); q b(k-1) , . . q b(1+2+...+(c-1) = q bc(c-1)/2 . , , Ai = (i - 1, a + i - 1) B j = (b + j - 1, j - 1). j = 1 q -
a+b . j q - b-i+ j

q ; j - 1, Ai B j , . . b - i + j . j - j - 1 . : P Pq (a, b, c) = q
-b
c(c-1) 2

det q

(b-i + j )( j -1)

a+b b-i+ j

c

.
i , j =1


.

( j -) q

b( j -1)

.

q , , . . . . (a, b, c) P Pq (a, b, c) = det q
( j -i )( j -1)

bc(c-1) 2

a+b b-i+ j

c

.
i , j =1

. . , i j : P Pq (a, b, c) = q
(12 +22 +...+(c-1)2 )

det q

-(i -1)( j -1)

a+b b-i+ j

c

.
i , j =1

§ . , , , . . , q -. , () . , -- , q - . . . (a, b, c) P Pq (a, b, c) = q
-a
c(c-1) 2

det

a+b+i-1 b+ j -1

c

.
q i , j =1

, . ( ), «» q -. . . (a, b, c)
b c

P Pq (a, b, c) =
i =1 j =1

[a + i + j - 1]q = [i + j - 1]q

a

b

c

i =1 j =1 k =1

[i + j + k - 1]q . [i + j + k - 2]q

.

. .


§ . .

. . . , ; . . [ ]. ; ., , [ , . I, ].

§ . .
, , q -
m+n n

m, n ,

:


(1 - q k )
k =1

-1

= lim

m , n

m+n . m

. (a, b, c) a, b . , c:


P Pq (c) =
i =1 j =1

[i + j + c - 1] . [i + j - 1]

: = i + j - 1. ,
[ + c] . []


P Pq (c) =
= 1

[ + c] []





=
= 1

1 - q+c 1 - q



.

c , , . . ().


P P (q ) =
= 1

(1 - q )- .


.

. . , . . [ ].


. . n = (n - 1, ..., 2, 1) -- . , n- Cn = n + 1 n . ( --). , , h, Cn C n +1 ... C n + h -1 C n +1 C n +2 ... C n + h det(Cn+i+ j -2)h, j =1 = det . . i . . . . C
n + h -1

1

2n

n

C

n+ h

...C

n +2 h -2

. . ,
1 i j n -1

.

i + j + 2h i+ j

. . () . . (0, 0), (0, 1), ..., (0, n - 1) (1 + n - 1, 1), (2 + n - 2, 1), ..., (n , n), n -- .





§ . . --
-- , , , ( - . .). «», , , , -- . , , - . A -- n â n, M -- . Mij n - 1, i - j - . M1n,1n -- n - 2, . , . . ( --). M·M
1n,1n

= M11 · M

nn

- M 1n · M n1 .

: -, ( -- , -- [ ]). --. A = (aij ) -- , M -- . , aij Mij , (-1)i+ j . (., , [ , § . ]) : ai1 Mi1 - ai2 Mi2 + ... + (-1)
n +1

ai n M i n = M .


.

, , « »: , j = i ai1 M j 1 - ai2 M j 2 + ... + (-1)
n +1

ai n M

jn

= 0.

M A = ((-1)
i+ j

M j i ),

. , , : A · A = M · E . , , A M , , A. A : , , . A:
M 11 - M 12 M 13 A= . . . n-1 (-1) M 0 0 ... 0 (-1)n-1 Mn 1 0 ... 0 (-1)n Mn2 0 1 ... 0 (-1)n+1 Mn . .. . . . 00...0 M nn
1

3



1n

A A. , M , -- , A:
A· A = Ma 0a . . . 0a
12 22

.

...a ...a ...a

1,n-1 2,n-1

n2

n,n-1

0 0 . . . M





.

det( A · A) = M 2 · M

1n,1n

,

M1n,1n -- , A .


§ . . --

, det A = M · M
1n,1n

.

A : det A = M11 · M
nn

- M 1n · M n1 .

-- .

§ . . --
--, « » ([ ]). , . , A = (aij ) -- n. n 1, 2, ..., n n 1 , 2 , ..., n . -- , : {1, ..., n} {1 , ..., n }, i (i ).
n

wt = sgn
i =1

a

i (i )

.

, ( ), . : {2, ..., n - 1} {2 ,

, 1- n-, , 1 n , -- ..., (n - 1) }, (i ) i . n -1

wt = sgn
i =2

a

i (i )

.

, , , .


.

A(n) -- (, ). : wt(, ) = wt · wt . -- A. , B(n) -- (, ), : {1, ..., n - 1} {1 , ..., (n - 1) } : {2, ..., n} {2 , ..., n }

(. . n n , ). (, ) . , C (n) -- (, ), , , : {2, ..., n} {2 , ..., n } : {1, ..., n - 1} {1 , ..., (n - 1) }.

(, ) wt(, ) = - wt · wt . -- B(n) C (n). T : A( n) B ( n) C ( n) . (, ) A(n). (mi wi ): m1 = n, w1 = (m1 ), m2 = -1 (w1 ), w2 = (m2 ) . . , wi mi mi+1 wi . wr , , . . wr = 1 n . (m1 , w1 ), (m2 , w2 ), ..., (mr , wr ) : mi wi , wi mi+1 . T ((, )) B(n) C (n). , . , T -- (. . ). . . . . n = 5. , -- . T (m5 , w4 ) (m3 , w5 ) , (m3 , w4 ) -- . .


§ . . « »

m

1

m

2

m

3

m

4

m

5

m

1

m

2

m

3

m

4

m

5

w1

w2

w3

w4

w5

w1

w2

w3

w4

w5

. . . T

T , . : B(n) C (n), T ( A(n)). . , B(n) C (n) S, wt S((, )) = - wt(, ). . . , . -- , .

§ . . « »
. ( ) [ ], , --. : M1n,1n = 0, -- : M=
1 M
1 n ,1 n

( M11 M

nn

- M 1n M n1 ) .

M n Mij n - 1 M1n,1n n - 2. . , n n - 1 n - 2 ( «»). , , «»,


.

, , . 3 â 3: a a a
11 21 31

a a a

12 22 32

a a a

13 23 33

=a

1
22

a a

11 21

a a

12 22

·

a a

22 32

a a

23 33

-

a a

12 22

a a

13 23

·

a a

21 31

a a

22 32

=

1 = a ((a11 a22 - a21 a12 )(a22 a 22

33

- a23 a32 ) -

- (a12 a23 - a13 a22 )(a12 a32 - a22 a31 )) = = a11 a22 a33 + a12 a23 a31 + a13 a21 a32 - a13 a22 a31 - a11 a23 a32 - a12 a21 a33 . , , - , , . ( , a22 = 0.) - , , -- , -- . . . -- « » (the Lewis Carroll identity). . .-. .; . , . ., . [ ] , « ».

§ . . -
, , 2 â 2. , - ( -- ). , - 2 â 2 det A = ab cd = a d + bc.



§ . . -

= -1 . - «- »: , A -- n, , - n - 1. det A = det ( A a a a a a a
1
22

1n,1n

)-1 (det A11 det A

nn

+ det A1n det An1 ).

. . - 3 â 3:
11 21 31 12 22 32

a a a a a
11 21

13 23 33

= a a a a a a a a a a a a a a

=a

12 22

·

22 32

23 33

+

12 22

13 23

·

21 31

22 32

=

= a ((a11 a22 + a21 a12 )(a22 a33 + a23 a32 ) + 22 + (a12 a23 + a13 a22 )(a12 a32 + a22 a31 )) = = a11 a22 a33 + a11 a23 a32 + a12 a21 a33 + 2 a12 a23 a31 + 2 a13 a21 a32 + + 3 a13 a22 a31 + 2 (1 + -1 )a12 a21 a23 a31 /a22 . - : -, - , -- , - n. -, n! , aij ; , , n , . . () i=1 ai(i) . () -- , . . . , = -1 () (-1) , . . «» . , - : 2 (1 + -1 )a12 a21 a23 a31 /a22 . = -1 , . , , -- . . Sn -- -

1


.

n, i - , (i )- , . , , , . -- . . . (- alternating sign matrix) -- n, · , -1 ; · ; · : , . , . . . . 3: 0 10 1 -1 1 . 0 10 «» 2 (1 + -1 )a12 a21 a23 a31 /a
22

- . . . , ( ). , , . . . (aij , akl ), (. . i > k j < l ). : (aij , akl ), i > k j < l , aij akl (. . , , -1 ). B I ( B). , N ( B).


§ . .

. .
0 1 0 0 0 1 B = 1 -1 0 0 1 -1 001 0 0 0 1 0 0 0 1 0 0 .

N ( B) = 2, I ( B) = 5: , «», -- «» ( !). - , . . . - A = (aij ) det A =
B A S M (n)



I ( B)

( 1 + - 1 )

N ( B) i, j

a i j ij ,

B

B, AS M (n) n. i, j aijij aij , , Bij B. , , B -- , i ai(i) . a12 a23 a31 a35 a42 a44 a53 /a32 a43 .
B

§ . .
: n? An . , A2 = 2, A3 = 7. An ; n 7 . n A
n

123

4

5

6

7

127 42 429 7436 218348 1 2 7 2 · 3 · 7 3 · 11 · 13 22 · 11 · 132 22 · 132 · 17 · 19

: An , .


.

, . An,k n, k - . , An,1 + An,2 + ... + An,n = An An,k : An,k = An,n-k+1 . . . , An,1 = An,n = An-1 . An,k , : 1 1 2 7 42 429 1287 105 2002 14 135 2002 3 14 105 1287 1 2 7 42 429

( : ). , , , . , -- , , , . , : 1 1 2 7 42 429
2 6 2 5 2 4 2 3 2 2

1
3 2

3
5 5

2
4 2

14
7 9

14
9 7

7
5 2

105
9 14

135
16 16

105
14 9

42
6 2

1287

2002

2002

1287

429

:


§ . .

, . ,
A A
5 ,1 5 ,2

=

2 , 5

A A

5 ,2 5 ,3

=

7 , 9

, ,
A A
6 ,2 6 ,2

=

2+5 9 = . 7+9 14

, « » 2/n n/2 . , , An,k / An,k+1 . , , , . . , , , : 1+1 1+1 1+1 1+1 1+1 4+5 3+4 6 + 10 2+3 3+6 4 + 10 1+2 1+3 1+4 1+5

. , . . .
A n ,k = An,k+1
n-1 n-2 + k -1 k -1 n-1 n-2 + n-k -1 n-k -1

.

. , alternating sign matrix conjecture. . ( ).
n -1

An =
j =0

(3 j + 1)! 1! · 4! · 7! · ... · (3n - 2)! = n! · (n + 1)! · ... · (2n - 1)! . (n + j )!

.

. .


.

, -- . . , , . (. [ ], [ ]). - . , An -- 1, 2, 7, 42, 429, ... -- .

§ . .
. , (a, b, c) a â b â c, . , P Pq (a, b, c) =
(a , b, c)

[ht + 1] , [ht ]

q = 1 P Pa
, b, c

=
(a , b, c)

ht + 1 ht

(, ht (i , j , k ) i + j + k - 2). : , , . : (a, a, c), x + y = 0, . . , 2 , . , . ; . . , -


§ . .

: S P Pq (a, a, c) =
(a,a,c)/
2

[||(ht + 1)] . [|| ht ]

( SPP «symmetric plane partitions».) . -, , 2 . -, || : , || 1, , 2 . , q = 1 : S P P ( a, a, c) =
(a,a,c)/
2

||(ht + 1) = || ht



(a,a,c)/

ht + 1 . ht
2

, . , a â a â a 120 240 . , . . , -- , ? CS P P (a, a, a), -- CS P Pq (a, a, a) ( «cyclically symmetric plane partitions»). - ( ). - - .; , 2 - . . ( ). CS P Pq (a, a, a) =
(a,a,a)/
3

[||(ht + 1)] . [|| ht ]

|| - , , , . q = 1 : CS P Pq (a, a, a) =
(a,a,a)/
3

||(ht + 1) = || ht



(a,a,a)/

ht + 1 . ht
3


.



. :

« , ». . , . . . [ ]. , , , , -- , , S3 . (- totally symmetric plane partitions, TSPP). , , . , TS P Pq (a, a, a) =
(a,a,a)/S
3

?

[||(ht + 1)] , [|| ht ]



S3 . ! q = 1 :
(a,a,a)/S

T S P P ( a, a, a) =
3

||(ht + 1) . || ht

. , q - . , , 2 - 3 - , : q - ||, , : TS P Pq (a, a, a) =
(a,a,a)/S
3

[ht + 1] . [ht ]

[ ].

. -


§ . .

, . . [ ] ( ). . ; [ ] . , .

§ . .
(a, b, c). -- . . , . , «» . , , . (complementar y) . , a, b, c, . . «», . , , (self-complementar y). , , . . . (2, 2, 2) (2, 2, 3). . . (totally symmetric self-complementary plane partition, TSSCPP) -- , (. . ) . .


.

. . . . . .

. . .



-- , ( ) . . . ; . , : 1, 2, 7, ... :

. . .


§ . .

, ! . . ( ). 2n T S S C P P (2 n ) =
1! · 4! · 7! · ... · (3n - 2)! . n! · (n + 1)! · ... · (2n - 1)!

. , . . [ ]. ( ) : 2n n. [ ]. ; ! , : , . -- , , -- , . , , «Proofs and confirmations» [ ], , . , (?) ; , , .


.

- , , . , , . , ( ) : , . , , . . . , . , . , ( [ ]), -- , --, . . .




. . ., . . , // . , . . .: , ( ). . . ., . ., . . . .: , . ( . . . . .: , ). . . . ? // . . . C. -- . . . . // . , . . .: , ( ). . C. . . .: , . . . . . .: , . . . . .: , . . . . , , , // . . . C. -- . . . . .: , . . Andrews G. Plane par titions, V: The T.S.S.C.P.P. conjecture // J. Combinatorial Th. (A). . Vol. ( ). P. -- . . Bressoud D. Proofs and confirmations: the stor y of the alternating signmatrix conjecture // MAA. . . Dodgson C. L. Condensation of determinants, being a new and brief method for computing their arithmetical values // Proceedings of the Royal Society -- --. of London. . Vol. . P. . Gessel I., Viennot X. Binomial determinants, paths, and hook length for--. mulae // Adv. Math. . Vol. . P. . Karlin S., McGregor J. G. The differential equations of bir th-and-death processes and the Stieltjes moment problem // Trans. Amer. Math. Soc. . -- . Vol. . P. . Karlin S., McGregor J. G. Coincidence probabilities // Pacific J. Math. . -- . Vol. . P. . Kuperberg G. Symmetries of plane par titions and permanent-determinant --. method // J. Combin. Theor y. Ser. A. . Vol. . P. . Kuperberg G. Another proof of the alternating-sign matrix conjecture // --. Internat. Math. Res. Notices. . Vol. . P.




. B. Lindstr¨m On the vector representations of induced matroids // Bull. o Lond. Math. Soc. . Vol. . P. -- . . Mills W. H., Robbins D. P., Rumsey H. Jr. Proof of the Macdonald conjecture // Invent. Math. . Vol. . P. -- . . O'Hara K. M. Unimodality of Gaussian coefficients: a constr uctive proof // J. Combin. Theor y. Ser. A. . Vol. . P. -- . . Pak I., Panova G. Strict unimodality of q-binomial coefficients. Preprint a r X iv: . . . . Stanley R. Symmetries of plane par titions // J. Combin. Theor y. Ser. A. . --. Vol. . P. . Stembridge J. The enumeration of totally symmetric plane par titions // -- Advances in Math. . Vol. . P. . . Zeilberger D. Kathy O'Hara's constr uctive proof of the unimodality of the -- Gaussian polynomials // Amer. Math. Monthly. . Vol. . P. . . Zeilberger D. Proof of the alternating sign matrix conjecture // Elect. J. of Combinatorics. . Vol. ( ). . Zeilberger D. Dodgson's determinant-evaluation r ule proved by two-timing men and women // Elect. J. of Combinatorics. . Vol. ( ).


,

. . . 60 â 90 / . . . . . , . . . . , , ., . . . ( )

­

­

« ». , , . , . .
« », ., . . . ( ) ­ ­ . E-mail: biblio@mccme.ru