3 p(3) , 3 +, 2 + 2, 2 + +, p(4) , 6, 7, 8 p(5), p(6), p(7), p(8). p(5) 7, p(6), p(7) 5, p(8) , + +

Transcription

1 k l. n n n pariion n p(n) p(n).. p() 2 2, , + 2 p(2) 2 3 3, 2 +, :00 7:00.

2 3 p(3) , 3 +, 2 + 2, 2 + +, p(4) , 6, 7, 8 p(5), p(6), p(7), p(8). p(5) 7, p(6), p(7) 5, p(8) , p(0) Young diagram 3 Ferrers diagram n k + k k r (k k 2 k r ) n k 2 k 2 r k r k + k k r 2 Alfred Young Norman Macleod Ferrers

3 .3., 2, n, k, l k n l k n l a n (k, l) k, l k, l a n a 0 (k, l) k 0 l 0 a 0 (k, 0), a n (k, 0) 0 (n > 0), a 0 (0, l), a n (0, l) 0 (n > 0) ().4 k l l k n.5. k l ,,,,, 3

4 6 a 0 (2, 2), a (2, 2), a 2 (2, 2) 2, a 3 (2, 2), a 4 (2, 2) n 5 a n (2, 2) 0 a n (k, l) (kl + ) a 0 (k, l), a (k, l), a 2 (k, l),, a kl (k, l) kl F (k, l; ) a n (k, l) n a 0 + a + a a kl kl (2) n0 n > kl a n (k, l) 0 F (k, l; ) {a n (k, l)} generaing funcion a n (k, l) n, k, l.2 F (k, l; ).6. () k 0 l 0 k l 2.5 F (0, l; ), F (k, 0; ) (3) F (2, 2; ) ( )( + 2 ).7. F (, l; ), F (k, ; ), F (3, 2; ). k a n (, l) (0 n l) a n (, l) 0 (n > l) F (, l; ) l l a n (k, ) (0 n k) a n (k, ) 0 (n > k) k F (k, ; ) k n k 3, l 2 l n n F (3, 2; ) n ( )( + 2 ) 4

5 .3 - kl F (k, l; ) a n (k, l) a 0 + a + + a kl n0 l k k 5, l F (k, l; ) l, k k l.. ( ) k + l F (k, l; ) (4) l ( ) k + l k+l C l n C m ( ) l n m ( ) n n! (5) m m!(n m)! i [i] i i i [] 0 < m < n m] [n] [n ] [] [m] [m ] [] [n m] [n m ] [] (6) 5

6 m 0 m n ] 0 n] (7) 4 Gaussian polynomial - binominal coefficien m]. [i] i (5) (6) [ ( ) n n (8) m] m.8. m m n ] [ ] n [n] [n ] [n 2] [2] [] n [] [n ] [n 2] [2] [] [n] [] n [ ] 4 [4] [3] [2] [] [4] [3] ( 4 )( 3 ) 2 [2] [] [2] [] [2] [] ( 2 )( ) ( )( + 2 ).9. n m [n] [m] [n] /[m] [n/m] [ ] [4] 4 + 2, [2] + [2] < m n m] m] [ ] n m n m m [ ] n + m, (9) m ] [ ] n +. (0) m 4 Johann Carl Friedrich Gauss, k j0 ( e 2π /k ) j 2 6

7 (9), (0) (8) ( ) ( ) ( ) n n n + () m m m. [k]! [k] [k ] [] [k + ]! [k + ] [k]! (9) [ ] [ ] n n + m m m [n ]! [m ]![n m]! + [n ]! m [m]![n m ]! [n ]! [m]![n m]! {[m] + m [n m] } [n ]! [m]![n m]! ( m ) + m ( n m ) [n ]! [m]![n m]! n [n ]! [m]![n m]! [n] m] (0).0.. m] [ ] 0. n n 0 0 [ ] [ ] [ ] n n n n,,, 0 n [ [ ] n n m n.0 (9) + m] [ ] m n ] m (7) m 0 n] n, m (0 m n) m] 7

8 k, l [ ] k + l F (k, l; ) l k n [ ] l k + l a n (k, l) n l. k 0 l 0 (3) (7) [ ] [ ] k l F (0, k; ), F (l, 0; ) 0 l k > 0, l > 0 k + l (2) k l [ ] [ ] + 2 F (, ; ) +, + (2) k + l > k l.7.8 [ ] + l F (, l; ) l, l [ ] k + F (k, ; ) k (2) k >, l > (2) F (k, l; ) F (k, l ; ) + l F (k, l; ) (3) l k.. 8

9 l k k k l l l k n a n (k, l ) l k k k l l l k n 2 a n l (k, l) a n (k, l) a n (k, l ) + a n l (k, l) n > k(l ) a n (k, l ) 0 m < 0 a m (k, l) 0 kl F (k, l; ) a n (k, l) n n0 9

10 kl (a n (k, l ) + a n l (k, l)) n n0 k(l ) n0 k(l ) n0 kl a n (k, l ) n + l a n l (k, l) n l nl (k )l a n (k, l ) n + l m0 F (k, l ; ) + l F (k, l; ) a m (k, l) m (3) (3).0 (3) F (k, l; ) F (k, l ; ) + l F (k, l; ) [ ] [ ] k + (l ) (k ) + l F (k, l ; ), F (k, l; ) l l [ ] k + l F (k, l; ) l [ ] k + l + l. l.0 (9) [ ] k + l F (k, l; ) l.3. k l 3 (3) 2-5 k l :00 7:00. 0

11 k l k n l a n (k, l) a 0 (k, l) + a 2 (k, l) + a 4 (k, l) k l ,,, 4 a n (k, l) kl F (k, l; ) a n (k, l) n a 0 + a + a a a a kl kl n0 kl F (k, l; ) a n (k, l) a 0 + a + a 2 + a 3 + a 4 + n0 kl F (k, l; ) ( ) n a n (k, l) a 0 a + a 2 a 3 + a 4 n0 2 ((F (k, l; ) + F (k, l; )) a 0 + a 2 + a 4 + a 6 + (4) + ( ) n 2 { n 0 n 6 Hook lengh formula laice pah mehod

12 2.2.2 k, l F (k, l; ) [ ] k + l F (k, l; ) l (4) [ 2. n ( ) m] n 7 m ( ) ( ) n n m m ( ) n m () ( ) ( ) ( ) n n n + m m m.0 m] 7 Blaise Pascal

13 [ ] n m m m] [ n ] m.0 (9) [ ] [ ] n n + m m m m].0 (9) ] m] m + ( ) m m ] (5) 3

14 m] m] m n, 0 l < n [ ] ( ) 2n n, 2m m [ ] 2n 0 2l + 0 m n [ ] 2n + 2m [ ] 2n + 2m + ( ) n m [ ] 0. n n 0 [ ] 0 [ ] 0 0 [ ] 0 [ ] ( ) 0 0 n > 0 (5) [ ] [ ] [ ] 2n 2n 2n + ( ) 2m 2m 2m 2m [ ] [ ] [ ] 2n 2n 2n + ( ) 2m+ 2m + 2m 2m + 4, [ ] 0

15 [ ] ( ) 2n n, 2m m [ ] 2n 2m [ ] 2n 2m + (5) [ ] 2n + 2m [ ] 2n + 2m + [ ] 2n 0, 2m [ ] 2n + 2m 0 + ( ) n m [ ] 2n 2m [ ] 2n 2m + ( ) ( ) ( ) n n n +, m m m ( ) ( ) n n 0. m m [ ] [ ] 2n 2n + ( ) 2m 2m 2m [ ] [ ] 2n + ( ) 2m+ 2n 2m 2m + [ ] 2n 2m ( ) n, m ( ) n, m [ ] 2n + 2m +, [ 2n ] 2m + ( ) n 0 ( ) n + 0 m m ( ) n m (4) k l {( ) ( )} k + l (k + l)/2 + k, l 2 l l/2 {( ) ( )} k + l (k + l )/2 + k l 2 l (l )/2 {( ) ( )} k + l (k + l )/2 + k l 2 l l/2 ( ) k + l k, l 2 l 5

16 k 3 l k, l k l k l l, k 80 l, k kl k l 3,,,,,,,,, n ( ) n ( + z) n z l (6) l l0 6

17 ( ) n + 0 ( ) n z + ( ) ( ) n n z z n + 2 n ( ) n z n. n - c, c 2,, c n 8 n c i c c 2 c n i 2.7. n ( + i z) i n ] l(l+)/2 z l. (7) l l0 ( + z)( + 2 z) ( + n z) ] 0 ] ] ] + z + 3 z n(n+)/2 z n. 2 n - (7) (8) (6) [ ] [ ] 2.8. n (7) n 2 0 [ ] 2 0, ( + z)( + 2 z) + ( + 2 )z + 3 z 2 [ ] 2 +, + ( + )z + 3 z 2 [ ] 2 (7) n 3, 4, 5 (7) n n 2.8 (7) 8 produc P π sum S 7

18 n > n n [ ] n ( + i z) k(k+)/2 z k k i k0 + n z n n ( + i z) ( + i z) ( + n z) i i ( n [ ] n k(k+)/2 k k0 n [ ] n k(k+)/2 k k0 ) z k ( + n z) z k + n k0 [ ] n k(k+)/2+n z k+ k z l l 0 k 0 z 0 [ ] n ] 0 0 l n 2 k n z n [ ] n ] n(n )/2+n n(n+)/2 n(n+)/2 n n l n k l 2 k l z l l(l+)/2 l ] + l(l )/2+n l ] ([ ] [ ] ) n n l(l+)/2 + n l l l.0 (0) ] l(l+)/2 l n ( + i z) i (7) n ] l(l+)/2 l l0 z l.2 8

19 2..2 n ( + i z) i ] F (n l, l; ) l n l(l+)/2 F (n l, l; )z l (8) l0 (8) n l l (8) x i i n ( + x i z) ( + x z)( + x 2 z) ( + x n z) i z l x i() x i(2) x i(l) ( i() < i(2) < < i(l) n) x i i x i() x i(2) x i(l) i(l) 2 i(l ) l i() x 2 x 3 x 5 x 7 l n l l.... l l n l n 7, l 4 ( + x )( + x 2 ) ( + x 7 ) z 4 x 2 x 3 x 5 x 7 9

20 z l l n l l l l(l + ) 2 n i ( + i z) z l l(l+)/2 F (n l, l; ) 2.0. n 3 ( + x z)( + x 2 z)( + x 3 z) + (x + x 2 + x 3 )z + (x x 2 + x x 3 + x 2 x 3 )z 2 + x x 2 x 3 z 3 l z x, x 2, x 3 x, x 2, x 3. 3 i ( + i z) z F (2, ; ) l 2 z 2 x x 2, x x 3, x 2 x 3 x x 2, x x 3, x 2 x 3. 3 i ( + i z) z 2 +2 F (, 2; ) z 3 x x 2 x 3 x x 2 x 3 3 i ( + i z) z F (3, 0; ) 20

21 2.. n 4, l 2 ( + x z)( + x 2 z)( + x 3 z)( + x 4 z) z x x 2, x x 3, x x 4, x 2 x 3, x 2 x 4, x 3 x 4. 4 i ( + i z) z 2 z +2 F (2, 2; ) 2.2. n 5 5 i ( + x iz) z 2, z 3 5 i ( + i z) z 2, z 3 z 3 F (3, 2; ), z 6 F (2, 3; ) 3 9 p(n) n0 p(n)n 3. n n p(n) p(0) n p(n) n p(n) :00 7:00. 2

22 p(n) p(0) + p() + p(2) 2 + p(3) 3 + p(4) p(n) n n p(n) p(n) n n0 k0 p(0) + p() + p(2) 2 + p(3) 3 + p(4) p(n) n + k (9) 2 3 k 3.2 a 0 + a + a a d d formal power series a 0 + a + a a n n + (20) n0 a n n p(n) n n0 n > d n a n 0 (20) 22

23 i , i0 (i + ) i , i0 2 i i i0 2 i0 a i i, i0 b i i a i i + i0 b i i i0 (a i + b i ) i, (2) ( a0 + a + a a a ) + ( b 0 + b + b b b ) i0 (a 0 + b 0 ) + (a + b ) + (a 2 + b 2 ) 2 + (a 3 + b 3 ) 3 + (a 4 + b 4 ) 4 + ( ) a i i i0 ( ) b i i i0 ( i ) a j b i j i, (22) i0 j0 ( a0 + a + a a a ) (b 0 + b + b b b ) a 0 b 0 + (a 0 b + a b 0 ) + (a 0 b 2 + a b + a 2 b 0 ) 2 + (a 0 b 3 + a b 2 + a 2 b + a 3 b 0 ) 3 + (a 0 b 4 + a b 3 + a 2 b 2 + a 3 b + a 4 b 0 ) 4 + n 3.4. ( ) ( ) i (i + ) i. i0 i0 23

24 . i ( ) ( ) i (i + ) i i0 i0 ( ) ( ) + ( + 2) + ( ) 2 + ( ) 3 + ( ) ( (i + )) i (i + )(i + 2) i. 2 i0 (i + )(i + 2) i ( ) ( α) α i i (23) ( ) ( α) α i i i0 i0 ( α) ( + α + α α ) + (α α) + (α 2 α α) 2 + (α 3 α α 2 ) (α i α α i ) i + (23) α α i i + α + α α i i0 i

25 k + k + 2k + 3k + 2i, i0 3i, i0 ki (24) 3.6. i0 ( + k ) ( + )( + 2 )( + 3 ) k m k ( + k ) m, 2, 3, 4 + +, ( + )( + 2 ) , ( + )( + 2 )( + 3 ) , ( + )( + 2 )( + 3 )( + 4 ) , ( + )( + 2 )( + 3 )( + 4 )( + 5 ) m, 2, 3, 4, 5 2 m 2, 3, 4, 5 3 m 3, 4, 5 4 m 4, 5 ( + 6 ), ( + 6 )( + 7 ), 5 m 5 m k ( + k ) 0,, 2, 3, 4, 5 m i m k ( + k ) i m i a i ( + k ) a i i k 25 i0

26 ( + k ) k f (), f 2 (), f 3 (), i m m k f k() i m m a i f k () a i i k 3.7. n k ( + k) m k ( + k) m(m + )/2 m i (9) k k > n k + k + 2k + 3k + k + n m k /( k ) n m n m k /( k ) n m i /( i ) (m n) n m n m k k 2 3 ( ) ( ) ( ) 26 4 m

27 ( ) ( + m + 2m + 3m + 4m + ) n p(n) k + k + 2k + 3k + rk ( k ) r r k k r.. 4 /( ), /( 2 ), /( 4 ) ( ) 4, ( 2 ) 2, ( 4 ) /( k() ) r()k() /( k(h) ) r(h)k(h) r()k() r(h)k(h) r() k() r(h) k(h) ( ) ( 3 ) ( 4 ) ( 6 ) ( ) 3 ( 3 ) 2 ( 4 ) ( 6 ) 27

28 9 n 3.8. n p(0) n /( ) p() n 2 2 /( ) 2 ( ) 2 /( 2 ) 2 ( 2 ) ( ) 2, ( 2 ) 2 2 p(2) 3 /( ) ( ) 3 /( ) ( ) /( 2 ) ( 2 ) /( 3 ) ( 3 ) ( ) 3, ( ) ( 2 ), ( 3 ) 3 3 p(3) 28

29 4 4 4 ( ) 4, ( ) 2 ( 2 ), ( ) ( 3 ) ( 2 ) 2, ( 4 ) p(4) , 6 p(5), p(6) m n m k /( k ) n n (9) n n p(n) (9) 3.4 n n q(n) n p odd (n) q(0) p odd (0) 3.0. n n

30 q() p odd (), q(2) p odd (2), q(3) p odd (3) 2, q(4) p odd (4) 2, q(5) p odd (5) 3, q(6) p odd (6) 4. n 6 q(n) p odd (n) 3.. n n q(n) p odd (n). (25). q(n) n n0 p odd (n) n 3. p odd (n) n n0 k q(n) n n0 n0 ( + k ) (26) k ( + )( + 2 )( + 3 ) 2k (27) 3 5 ( )( )( ) ( + k ) k + i ( 2i )/( i ) k ( + )( + 2 )( + 3 )( + 4 )( + 5 ) k (28)

31 3 5 ( 6 )( 8 )( 0 ) n 0,, 2, 3, 4, 5 ( + )( + 2 )( + 3 )( + 4 )( + 5 ) n 3 5 n ( + i ) n i i 2i n n m n m ( + )( + 2 )( + 3 ) ( + m ) m 3 m 3 ( m m+ )( m+3 ) ( 2m ) i ( + i ) n i /( 2i ) n (28) ( + i ) i i 2i i i ( 2i ) i ( i ) 2i i 3. n n i ( i ) p(n) :00 7:00. 3

32 4. 4. p(n) n n0 i i ( i ) 50 i ( i ) i ( i ) i 0,, 0,,,,,,,,,,, k(3k )/2 2 k(3k + )/2 4.. ( i ) + ( ) k k(3k )/2 + i k ( ) l l(3l+)/2 (29) l 4. k(3k )/2, 2, 3, 4, Leonhard Euler

33 2 k k(3k )/2, 2, k(k + )/2 (k ) i ( i ) i ( + i ) 3. n q(n) ( + i ) q(n) n i 33 n0

34 + i i i i ( + i ) , 2,, i ( i ) i + ( i ) i 3 ( ), ( ) ( ) + ( )

35 ,, 6 ( ) 2 ( ) 3 ( ) ( ) 2 4 ( ) ( ) 2 5 ( ) ( ) 2 ( ) 2 6 ( ) ( ) 2 ( ) 2 ( ) 3,, ( ) + ( ) ( ) + ( ) ( ) + ( ) 2 + ( ) 2 6 ( ) + ( ) 2 + ( ) 2 + ( ) 3 0 r ( ) r 4.3. n q + (n) n q (n) ( i ) i (q + (n) q (n)) n (30) n0 q + (0), q (0)

36 4.4. n q + (n) q (n) { ( ) k n k n k(3k ± )/2 0 (3) (a) n k n k(3k ± )/2 q + (n) q (n) +, (b) n k n k(3k ± )/2 q + (n) q (n), (c) q + (n) q (n). n n (a) n k(3k ± ) k (b) n k(3k ± ) k (c) a, b m + (m ) + + (m a + ) +((m a) ) + + b. }{{} a m b a b a 2, b 3 a, b 36

37 a < b a a (m ) + (m 2) + + (m a) +((m a) ) + + b + a }{{} a a b b b b {}}{ (m + ) + m + + (m b + 2) +(m b) + + (m a + ) }{{} a + ((m a) ) + a < b a a b b a 2, b a 5, b a 3, b a, b a, b 2 37

38 4.6. n a b n 2 2 a, b 2 + n 3 n 4 n a 2, b 2 n 6, n 7, 38

39 4 + 3 a 2, b n 8, 9, 0,, 2 () (2) (),,, (2),,, (2k ) + (2k) + + (k + ) + k }{{} k 2k k.... k a b k (2k) + (2k ) + + (k + 2) + }{{} k (2k) + (2k ) + + (k + 2) + (k + ) }{{} k 39

40 2k k.... k + a k, b k + (2k ) + (2k 2) + + (k + ) + k +k }{{} k p(n) n n0 4. i0 i + ( ) k k(3k )/2 + k ( ) l l(3l+)/2 l ( i ) i 2 (p(0) + p() + p(2) 2 + p(3) 3 + p(4) 4 + ) ( ) n p(n) p(n ) p(n 2) + p(n 5) + p(n 7) p(n 2) p(n 5) + 0 p(n) 40

41 4.8. p(n) k ( ) k p (n 2 ) k(3k ) + l ( ) l p (n 2 ) l(3l + ) (32) k, l n k(3k )/2, n l(3l + )/2 p(n) 4.9. p(0) p() p(0), p(2) p() + p(0) 2, p(3) p(2) + p() 3, p(4) p(3) + p(2) 5, p(5) p(4) + p(3) p(0) 7, p(6) p(5) + p(4) p(), p(7) p(6) + p(5) p(2) p(0) 5, p(8) p(7) + p(6) p(3) p() p(9), p(0),, p(5) G. E. Andrews and K. Eriksson, Ineger Pariions, Cambridge Univ. Press, Andrews Eriksson 2 G. E. Andrews, The Theory of Pariions, Cambridge Univ. Press, 998. Andrews 30 4

42 R. P. Sanley, Enumeraive Combinaorics, Vol., Cambridge Univ. Press, 996. I R. P. Sanley, Enumeraive Combinaorics, Vol. 2, Cambridge Univ. Press,