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ɇȿɄɈɌɈɊɕȿ ɋȼȿȾȿɇɂə ɂɁ ȺɅȽȿȻɊɕ ɂ ɄɈɆȻɂɇȺɌɈɊɂɄɂ
ɉɨɥɭɱɟɧɧɵɟ ɮɨɪɦɭɥɵ ɫɨɜɩɚɥɢ ɫ ɮɨɪɦɭɥɚɦɢ (14). Ɉɬɫɸɞɚ ɩɨɥɭɱɚɟɦ ɢɫɤɨɦɵɣ ɡɚɤɨɧ ɪɚɫɩɪɟɞɟɥɟɧɢɹ:
S 1 2 … K … n
1
P
n
)2 ,(nns
…
!
) ,(nkns
…
!
!1n
ɝɞɟ ɫɥɭɱɚɣɧɚɹ ɜɟɥɢɱɢɧɚ S – ɱɢɫɥɨ ɩɨɹɜɥɟɧɢɣ ɨɩɟɪɚɰɢɢ ɩɪɢɫɜɚɢɜɚɧɢɹ
ɜ ɚɥɝɨɪɢɬɦɟ 6, ɚ Ɋ
ɩɟɪɟɫɬɚɧɨɜɤɢ ɱɢɫɥɨ ɨɩɟɪɚɰɢɣ « = » ɩɪɢ ɨɛɪɚɛɨɬɤɟ ɚɥɝɨɪɢɬɦɨɦ 6 ɛɭɞɟɬ ɪɚɜ-
S
n
= Ɋ(S = k) – ɜɟɪɨɹɬɧɨɫɬɶ ɬɨɝɨ, ɱɬɨ ɞɥɹ ɫɥɭɱɚɣɧɨ ɜɡɹɬɨɣ ɢɡ
k
ɧɨ k, k = 1, 2, …, n.
ɋɪɟɞɧɟɟ ɱɢɫɥɨ ɜɵɩɨɥɧɟɧɢɣ ɭɤɚɡɚɧɧɨɣ ɤɨɦɚɧɞɵ – ɦɚɬɟɦɚɬɢɱɟɫɤɨɟ ɨɠɢɞɚɧɢɟ MS ɧɚɣɞɟɦ, ɩɨɥɶɡɭɹɫɶ ɮɨɪɦɭɥɨɣ (15)
n
k
1
n
!
n
1
kSPkMS
))((
n
!
k
c
nxxx
)1)...(1(
x
1
knsk
n
1
n
!
k
1
n
1
§
|)),(|(
¨
¦¦¦
n
!
k
011
©
kx
1
nxxx
c
·
k
xkns
|),(|
¸
¹
x
1
n
n
)1)...(1(
k
x
1
n
11!!
.
¦¦¦
kkn
k
11
ɂɬɚɤ, MS ɪɚɜɧɨ ɝɚɪɦɨɧɢɱɟɫɤɨɦɭ ɱɢɫɥɭ, ɬɨ ɟɫɬɶ ɦɟɧɶɲɟ lnn + 0,5.
Ɂɚɞɚɱɚ ɨ ɫɥɨɠɧɨɫɬɢ ɚɥɝɨɪɢɬɦɚ 6 ɪɟɲɟɧɚ.
§ 8. ɑɢɫɥɚ ɋɬɢɪɥɢɧɝɚ ɜɬɨɪɨɝɨ ɪɨɞɚ
ɇɟɩɭɫɬɵɟ ɢ ɧɟɩɟɪɟɫɟɤɚɸɳɢɟɫɹ ɩɨɞɦɧɨɠɟɫɬɜɚ ɦɧɨɠɟɫɬɜɚ ɛɭɞɟɦ ɧɚɡɵɜɚɬɶ ɛɥɨɤɚɦɢ ɦɧɨɠɟɫɬɜɚ. ɉɪɟɞɫɬɚɜɥɟɧɢɹ ɦɧɨɠɟɫɬɜɚ ɨɛɴɟɞɢɧɟɧɢɟɦ ɛɥɨɤɨɜ –
ɪɚɡɛɢɟɧɢɟɦ ɦɧɨɠɟɫɬɜɚ ɧɚ ɛɥɨɤɢ. ɇɚɩɪɢɦɟɪ,
ɞɨɟ ɪɚɡɛɢɟɧɢɟ n-ɷɥɟɦɟɧɬɧɨɝɨ ɦɧɨɠɟɫɬɜɚ ɢɦɟɟɬ ɬɢɩ:
ɞɟɪɠɢɬ Ȝ
i-ɷɥɟɦɟɧɬɧɵɯ ɛɥɨɤɨɜ. ɇɚɩɪɢɦɟɪ, ɪɚɡɛɢɟɧɢɟ 7-ɷɥɟɦɟɧɬɧɨɝɨ ɦɧɨɠɟɫɬ-
i
ɜɚ {1, 2, 3, 4, 5, 6, 7} = {1, 7}{2, 3}{4, 5, 6}ɢɦɟɟɬ ɬɢɩ 2
m
,
k
k
1
OO
231
BBBA
O
21
n
...21
.
, ij. Ʉɚɠ-
ji
n
, ɟɫɥɢ ɨɧɨ ɫɨ-
Ɉɱɟɜɢɞɧɨ, ɱɬɨ ɟɫɥɢ ɪɚɡɛɢɟɧɢɟ n-ɷɥɟɦɟɧɬɧɨɝɨ ɦɧɨɠɟɫɬɜɚ ɢɦɟɟɬ ɬɢɩ:
O
OO
n
21
n
...21
, ɬɨ
+ 2Ȝ2 + … + nȜn = n.
Ȝ
1
ɇɚɣɞɟɦ ɱɢɫɥɨ ɪɚɡɛɢɟɧɢɣ n-ɷɥɟɦɟɧɬɧɨɝɨ ɦɧɨɠɟɫɬɜɚ Ⱥ ɧɚ m ɛɥɨɤɨɜ
, ȼ2, …, ȼm, ɫɨɫɬɨɹɳɢɯ ɢɡ ɡɚɞɚɧɧɨɝɨ ɱɢɫɥɚ ɷɥɟɦɟɧɬɨɜ, ɬɨ ɟɫɬɶ |ȼ1| = k1,
ȼ
1
| = k2, …, |ȼm| = km. Ɉɱɟɜɢɞɧɨ, ɱɬɨ k1 + k2 + …+ km = n. ɗɬɨ ɪɚɡɛɢɟɧɢɟ ɦɨɠɧɨ
|ȼ
2
ɩɨɥɭɱɢɬɶ ɬɚɤ. ȼɨɡɶɦɟɦ ɩɪɨɢɡɜɨɥɶɧɨɟ k
ɫɬɜɚ Ⱥ. ɗɬɨ ɦɨɠɧɨ ɫɞɟɥɚɬɶ
k
1
C
n
-ɷɥɟɦɟɧɬɧɨɟ ɩɨɞɦɧɨɠɟɫɬɜɨ ɦɧɨɠɟ-
1
ɫɩɨɫɨɛɚɦɢ, ɡɚɬɟɦ ɢɡ ɨɫɬɚɜɲɟɝɨɫɹ (n – k1)-
61
ɉɊɂȼȺɅɈȼ Ⱥ. Ⱥ. ɗɅȿɆȿɇɌɕ ɉɊɈȽɊȺɆɆɂɊɈȼȺɇɂə ɉɊɂ Ɋȿɒȿɇɂɂ ɆȺɌȿɆȺɌɂɑȿɋɄɂɏ ɁȺȾȺɑ
k
2
ɷɥɟɦɟɧɬɧɨɝɨ ɦɧɨɠɟɫɬɜɚ
C
ɫɩɨɫɨɛɚɦɢ ɜɵɛɟɪɟɦ k2-ɷɥɟɦɟɧɬɧɨɟ ɩɨɞɦɧɨɠɟɫɬ-
kn
1
ɜɨ ɢ ɬ.ɞ. ɉɨɥɶɡɭɹɫɶ ɨɫɧɨɜɧɵɦ ɩɪɚɜɢɥɨɦ ɤɨɦɛɢɧɚɬɨɪɢɤɢ, ɧɚɣɞɟɦ ɱɢɫɥɨ ɧɚɲɢɯ
ɪɚɡɛɢɟɧɢɣ:
!
k
3
211
k
m
...
CCCC
kknkknkn
21
)!(
kkn
21
kkknk
3213
...
kkkn
m
121
...
)!(!
n
21
)!(!
knk
11
)!...(
kkkn
m
)!...(!
kknk
)!(
kn
1
)!(!
kknk
212
!
n
!!...!
kkk
211
mmm
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɦɵ ɞɨɤɚɡɚɥɢ ɫɥɟɞɭɸɳɭɸ ɬɟɨɪɟɦɭ.
Ɍɟɨɪɟɦɚ 14. ɑɢɫɥɨ ɪɚɡɛɢɟɧɢɣ n-ɷɥɟɦɟɧɬɧɨɝɨ ɦɧɨɠɟɫɬɜɚ Ⱥ ɧɚ m ɛɥɨɤɨɜ
, ȼ2, … , ȼm, ɤɚɠɞɵɣ ɢɡ ɤɨɬɨɪɵɯ ɫɨɞɟɪɠɢɬ ɩɨ k1, k2, … , km, ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ,
ȼ
1
ɪɚɜɧɨ ɋ
ɑɢɫɥɚ ɋ
ɦɢ. ɂ ɜɨɬ ɩɨɱɟɦɭ. ȿɫɥɢ ɪɚɫɤɪɵɬɶ ɫɤɨɛɤɢ ɜ ɜɵɪɚɠɟɧɢɢ (ɚ
ɩɨɥɭɱɢɦ ɫɭɦɦɭ m
ɜ ɤɚɠɞɨɦ ɢɡ ɧɢɯ ɚ
ɪɚɡ. Ɍɨɝɞɚ ɱɢɫɥɨ ɫɥɚɝɚɟɦɵɯ ɜɢɞɚ
k
m
ɋ
n(k1, k2
{1, 2, … , n} ɜ ɜɢɞɟ ɨɛɴɟɞɢɧɟɧɢɹ m ɩɨɞɦɧɨɠɟɫɬɜ ȼ
(i = 1, 2, … m) ɷɥɟɦɟɧɬɨɜ ɤɚɠɞɵɣ.
ɩɨ k
i
, ... , km).
n(k1, k2
, ... , km) ɟɳɟ ɧɚɡɵɜɚɸɬ ɩɨɥɢɧɨɦɢɚɥɶɧɵɦɢ ɤɨɷɮɮɢɰɢɟɧɬɚ-
n(k1, k2
n
ɫɥɚɝɚɟɦɵɯ (Ⱦɨɤɚɠɢɬɟ!) ɜɢɞɚ
ɜɫɬɪɟɱɚɟɬɫɹ k1 ɪɚɡ, ɚ2 ɜɫɬɪɟɱɚɟɬɫɹ k2 ɪɚɡ, …, ɚm ɜɫɬɪɟɱɚɟɬɫɹ
1
k
kk
m
21
,...
aaa
ɩɨ ɬɟɨɪɟɦɟ 8, ɛɭɞɟɬ ɪɚɜɧɨ
m
21
+ ɚ2 + … + ɚm)n, ɬɨ
1
aaaaa ...
3121
n
m
ɢ ɩɭɫɬɶ
, ... , km) – ɱɢɫɥɭ ɫɩɨɫɨɛɨɜ ɩɪɟɞɫɬɚɜɥɟɧɢɹ n-ɷɥɟɦɟɧɬɧɨɝɨ ɦɧɨɠɟɫɬɜɚ
, ȼ2, … , ȼm, ɫɨɞɟɪɠɚɳɢɯ
1
ɉɪɢ m = 2 ɩɨɥɭɱɚɟɦ ɮɨɪɦɭɥɭ ɛɢɧɨɦɚ ɇɶɸɬɨɧɚ ɢ ɛɢɧɨɦɢɚɥɶɧɵɟ ɤɨɷɮɮɢɰɢɟɧɬɵ.
ɑɢɫɥɨ ɠɟ ɪɚɡɛɢɟɧɢɣ n-ɷɥɟɦɟɧɬɧɨɝɨ ɦɧɨɠɟɫɬɜɚ Ⱥ ɧɚ m ɩɪɨɢɡɜɨɥɶɧɵɯ
ɛɥɨɤɨɜ ɧɚɡɵɜɚɟɬɫɹ ɱɢɫɥɨɦ ɋɬɢɪɥɢɧɝɚ 2-ɝɨ ɪɨɞɚ ɢ ɨɛɨɡɧɚɱɚɟɬɫɹ S(n, m). ɇɚɩɪɢɦɟɪ, S(4, 2) = 7: {1, 2, 3}{4}; {1, 2, 4}{3}; {1, 3, 4}{2}; {1, 2}{3, 4};
{1, 3}{2, 3}; {1, 4}{2, 3}; {1}{2, 3, 4}.
ɋ ɩɨɦɨɳɶɸ ɱɢɫɟɥ ɋɬɢɪɥɢɧɝɚ
ɫɸɪɴɟɤɬɢɜɧɵɯ ɮɭɧɤɰɢɣ
ɜɬɨɪɨɝɨ ɪɨɞɚ ɦɨɠɧɨ ɧɚɣɬɢ ɱɢɫɥɨ t
ɧɚ
BAf
o:
, ɝɞɟ |A| = n, |ȼ| = m ɢ m n > 0 (ɱɢɫɥɚ t
n,m –
n,m
ɬɚɤɠɟ ɧɚɡɵɜɚɸɬ ɱɢɫɥɚɦɢ ɋɬɢɪɥɢɧɝɚ ɩɟɪɜɨɝɨ ɪɨɞɚ, ɧɨ ɷɬɨ ɧɟ s(n, m) ɢ |s(n, m)|).
Ⱦɨɤɚɠɟɦ, ɱɬɨ
= m!S(n,m) (16)
t
n,m
Ɍɚɤ ɮɭɧɤɰɢɹ f ɫɸɪɴɟɤɬɢɜɧɚ, ɬɨ ɞɥɹ ɜɫɟɯ bB ɦɧɨɠɟɫɬɜɚ (ɩɪɨɨɛɪɚɡɵ
ɷɥɟɦɟɧɬɨɜ bB ɮɭɧɤɰɢɢ f ) f
ɦɧɨɠɟɫɬɜɚ ɧɟ ɩɟɪɟɫɟɤɚɸɬɫɹ. ȼ ɫɚɦɨɦ ɞɟɥɟ, ɟɫɥɢ ɫɭɳɟɫɬɜɭɟɬ aA ɬɚɤɨɣ, ɱɬɨ
-1
(b1) ɢ af-1(b2) ɩɪɢ ɧɟɤɨɬɨɪɵɯ b1 b2, ɬɨ ɷɬɨ ɛɭɞɟɬ ɩɪɨɬɢɜɨɪɟɱɢɬɶ ɨɩɪɟɞɟ-
af
ɥɟɧɢɸ ɮɭɧɤɰɢɢ. Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɮɭɧɤɰɢɹ f ɡɚɞɚɟɬ ɪɚɡɛɢɟɧɢɟ Ⱥ ɧɚ m ɛɥɨɤɨɜ:
N(f) = {f
)(
Bb
1
bfA
. Ɍɚɤɢɟ ɪɚɡɛɢɟɧɢɹ ɧɚɡɵɜɚɸɬ ɹɞɪɨɦ ɮɭɧɤɰɢɢ f ɢ ɨɛɨɡɧɚɱɚɸɬ
–1
(b) : bB}. Ɉɛɪɚɬɧɨ, ɤɚɠɞɨɦɭ ɪɚɡɛɢɟɧɢɸ {B1, B1,…, Bm} ɦɧɨɠɟɫɬ-
-1
(b) = {aA : f(a) = b} ɧɟ ɩɭɫɬɵ. Ʉɪɨɦɟ ɬɨɝɨ, ɷɬɢ
62
ɇȿɄɈɌɈɊɕȿ ɋȼȿȾȿɇɂə ɂɁ ȺɅȽȿȻɊɕ ɂ ɄɈɆȻɂɇȺɌɈɊɂɄɂ
ɜɚ Ⱥ ɧɚ m ɛɥɨɤɨɜ ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɪɨɜɧɨ m! ɮɭɧɤɰɢɣ ɫ ɹɞɪɨɦ N(f) = {B1, B1, …,
}. Ɋɚɜɟɧɫɬɜɨ (16) ɞɨɤɚɡɚɧɨ.
B
m
ɇɚ ɩɪɢɦɟɪɟ ɬɪɟɭɝɨɥɶɧɢɤɚ ɉɚɫɤɚɥɹ, ɫ ɩɨɦɨɳɶɸ ɤɨɬɨɪɨɝɨ ɥɟɝɤɨ ɧɚɯɨɞɢɬɶ
ɱɢɫɥɚ ɫɨɱɟɬɚɧɢɣ, ɜɵɜɟɞɟɦ ɞɥɹ ɱɢɫɟɥ ɋɬɢɪɥɢɧɝɚ 2-ɝɨ ɪɨɞɚ ɪɟɤɭɪɪɟɧɬɧɵɟ ɮɨɪɦɭɥɵ. ɉɟɪɜɨɣ ɮɨɪɦɭɥɨɣ ɛɭɞɟɬ ɫɥɟɞɭɸɳɟɟ ɫɨɨɬɧɨɲɟɧɢɟ:
S(n, m) = S(n – 1, m – 1) + mS(n – 1, m), (17)
ɝɞɟ ɨɱɟɜɢɞɧɨ, ɱɬɨ S(n, n) = 1 ɞɥɹ n 0 ɢ S(n, 0) = 0 ɩɪɢ n
> 0.
Ⱦɥɹ ɞɨɤɚɡɚɬɟɥɶɫɬɜɚ ɮɨɪɦɭɥɵ (17) ɦɧɨɠɟɫɬɜɨ ɜɫɟɯ ɪɚɡɛɢɟɧɢɣ ɦɧɨɠɟɫɬɜɚ
{1, 2, …, n} ɧɚ m ɛɥɨɤɨɜ ɪɚɡɨɛɶɟɦ ɧɚ ɞɜɚ ɤɥɚɫɫɚ. ɉɟɪɜɵɣ ɢɡ ɧɢɯ ɫɨɞɟɪɠɢɬ ɨɞ-
ɧɨɷɥɟɦɟɧɬɧɵɣ ɛɥɨɤ {n}, ɚ ɜɬɨɪɨɣ ɧɟ ɫɨɞɟɪɠɢɬ ɬɚɤɨɝɨ ɛɥɨɤɚ, ɬɨ ɟɫɬɶ n ɹɜɥɹɟɬɫɹ ɷɥɟɦɟɧɬɨɦ ɧɟɤɨɬɨɪɨɝɨ ɛɥɨɤɚ. ɗɬɢ ɤɥɚɫɫɵ, ɨɱɟɜɢɞɧɨ, ɧɟ ɩɟɪɟɫɟɤɚɸɬɫɹ, ɩɪɢɱɟɦ ɩɟɪɜɵɣ ɫɨɫɬɨɢɬ ɢɡ S(n – 1, m –
1) ɪɚɡɛɢɟɧɢɣ ɦɧɨɠɟɫɬɜɚ {1, 2, …, n – 1}
ɧɚ (m – 1) ɛɥɨɤ, ɚ ɜɬɨɪɨɣ ɢɡ mS(n – 1, m), ɝɞɟ ɜ ɛɥɨɤɢ ɪɚɡɛɢɟɧɢɣ ɦɧɨɠɟɫɬɜɚ
{1, 2, …, n – 1} ɧɚ m ɛɥɨɤɨɜ ɩɨɨɱɟɪɟɞɧɨ ɞɨɛɚɜɥɹɟɬɫɹ ɷɥɟɦɟɧɬ n. Ɋɚɜɟɧɫɬɜɨ
(17) ɞɨɤɚɡɚɧɨ.
Ⱥ ɜɨɬ ɞɪɭɝɚɹ, ɬɨɠɟ ɨɱɟɧɶ ɩɨɥɟɡɧɚɹ, ɪɟɤɭɪɪɟɧɬɧɚɹ ɮɨɪɦɭɥɚ:
n
¦
1
i
1
n
1
mi
miSCmnS
.)1 ,(),(
(18)
Ⱦɨɤɚɡɚɬɟɥɶɫɬɜɨ. ɉɪɢ m = 1 ɮɨɪɦɭɥɚ ɨɱɟɜɢɞɧɚ. ɉɭɫɬɶ m > 1
ɢ Ⱥ = {1, 2, …, n}. Ɇɧɨɠɟɫɬɜɨ ɜɫɟɯ ɪɚɡɛɢɟɧɢɣ S(n, m) ɦɧɨɠɟɫɬɜɚ
Ⱥ ɪɚɫɩɚɞɚɟɬɫɹ ɧɚ ɪɚɡɥɢɱɧɵɟ ɤɥɚɫɫɵ, ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɪɚɡɧɵɦ ɩɨɞɦɧɨɠɟɫɬɜɚɦ Ⱥ, ɤɨɬɨɪɵɟ ɹɜɥɹɸɬɫɹ ɛɥɨɤɚɦɢ, ɫɨɞɟɪɠɚɳɢɦɢ ɷɥɟɦɟɧɬ n. Ⱦɥɹ ɤɚɠɞɨɝɨ kɷɥɟɦɟɧɬɧɨɝɨ ɩɨɞɦɧɨɠɟɫɬɜɚ ȼȺ, ɫɨɞɟɪɠɚɳɟɝɨ ɷɥɟɦɟɧɬ n, ɫɭɳɟɫɬɜɭɟɬ ɪɨɜɧɨ
S(n – k, m – 1) ɪɚɡɛɢɟɧɢɣ
ɦɧɨɠɟɫɬɜɚ Ⱥ ɧɚ m ɛɥɨɤɨɜ, ɫɨɞɟɪɠɚɳɢɯ ȼ ɜ ɤɚɱɟɫɬɜɟ
ɛɥɨɤɚ. k-ɷɥɟɦɟɧɬɧɨɟ ɩɨɞɦɧɨɠɟɫɬɜɨ ȼȺ, ɫɨɞɟɪɠɚɳɟɟ ɷɥɟɦɟɧɬ n, ɦɨɠɧɨ ɜɵɛɪɚɬɶ
ɫɩɨɫɨɛɚɦɢ. Ɉɬɫɸɞɚ,
n
11k
C
)1(
mn
1
k
1
n
1
k
)1(
mn
kn
1
n
1
k
1
n
k
¦¦¦
1
n
1
mk
.)1,()1,()1,(),(
kkSCmknSCmknSCmnS
Ɋɚɜɟɧɫɬɜɨ (18) ɞɨɤɚɡɚɧɨ.
ɉɪɢɜɟɞɟɦ ɟɳɟ ɨɞɢɧ ɮɚɤɬ, ɫɜɹɡɚɧɧɵɣ ɫ ɱɢɫɥɚɦɢ ɋɬɢɪɥɢɧɝɚ 2-ɝɨ ɪɨɞɚ.
Ⱦɥɹ ɷɬɨɝɨ ɩɪɢɜɟɞɟɦ ɨɛɨɛɳɟɧɢɟ ɱɢɫɥɚ ɜɡɚɢɦɧɨ ɨɞɧɨɡɧɚɱɧɵɯ ɮɭɧɤɰɢɣ,
ɚ ɢɦɟɧɧɨ:
= x(x – 1)(x – 2)…(x – n + 1), [x]0 = 1, [x]1 = x, …
[x]
n
ɧɚɡɵɜɚɟɬɫɹ ɨɛɨɛɳɟɧɧɨɣ ɫɬɟɩɟɧɶɸ ɯ ɢ ɹɜɥɹɟɬɫɹ ɦɧɨɝɨɱɥɟɧɨɦ ɫɬɟɩɟɧɢ n.
[x]
n
63
ɉɊɂȼȺɅɈȼ Ⱥ. Ⱥ. ɗɅȿɆȿɇɌɕ ɉɊɈȽɊȺɆɆɂɊɈȼȺɇɂə ɉɊɂ Ɋȿɒȿɇɂɂ ɆȺɌȿɆȺɌɂɑȿɋɄɂɏ ɁȺȾȺɑ
Ⱦɨɤɚɠɢɬɟ, ɱɬɨ ɥɸɛɨɣ ɦɧɨɝɨɱɥɟɧ ɫɬɟɩɟɧɢ n ɦɨɠɧɨ ɜɵɪɚɡɢɬɶ ɱɟɪɟɡ ɦɧɨ-
ɝɨɱɥɟɧɵ [x]
, ɬɨ ɟɫɬɶ
n
n
k
k
k
0
10
n
xbxpbbbxaxp
][)(:,...,,!)(
¦¦
k
.
kkn
0
Ɍɟɨɪɟɦɚ 15. Ⱦɥɹ ɤɚɠɞɨɝɨ n 0
n
n
¦
k
0
xknSx
])[,(
. (19)
k
Ⱦɨɤɚɡɚɬɟɥɶɫɬɜɨ. ɉɭɫɬɶ ɯ – ɧɚɬɭɪɚɥɶɧɨɟ ɱɢɫɥɨ. ɉɨɥɶɡɭɹɫɶ ɨɫɧɨɜɧɵɦ
ɩɪɚɜɢɥɨɦ ɤɨɦɛɢɧɚɬɨɪɢɤɢ, ɥɟɝɤɨ ɩɨɤɚɡɚɬɶ, ɱɬɨ ɱɢɫɥɨ ɜɫɟɯ ɮɭɧɤɰɢɣ f: Ⱥ o ȼ,
ɝɞɟ |A| = n, |ȼ| = ɯ ɪɚɜɧɨ ɯ
n
. ɋ ɞɪɭɝɨɣ ɫɬɨɪɨɧɵ, ɤɚɠɞɚɹ ɮɭɧɤɰɢɹ f ɹɜɥɹɟɬɫɹ
ɫɸɪɴɟɤɰɢɟɣ f : Ⱥof(Ⱥ). Ɉɬɫɸɞɚ, ɞɥɹ ɥɸɛɨɝɨ Y ȼ, ɬɚɤɨɝɨ, ɱɬɨ |Y| = k, ɱɢɫɥɨ
ɜɫɟɯ ɮɭɧɤɰɢɣ f : Ⱥ o Y, ɝɞɟ Y = f(Ⱥ), ɜ ɫɢɥɭ (16) ɪɚɜɧɨ t
k-ɷɥɟɦɟɧɬɧɨɟ ɩɨɞɦɧɨɠɟɫɬɜɨ Y ɦɧɨɠɟɫɬɜɚ ȼ ɦɨɠɧɨ ɜɵɛɪɚɬɶ
Ɉɬɫɸɞɚ ɢ ɬɨ, ɱɬɨ S(n, k) = 0 ɞɥɹ k > n ɢ [x]
x
n
k
x
k
x
k
kknSCkknSx
!),(!),(
= 0 ɞɥɹ k < x, ɢɦɟɟɦ
k
x
x
!
kxk
)!(!
k
= k!S(n,k). Ⱦɚɥɟɟ,
n,k
k
C
ɫɩɨɫɨɛɚɦɢ.
n
n
¦¦¦¦
k
k
0000
xknSxknS
])[,(])[,(
k
ɢ ɪɚɜɟɧɫɬɜɨ (19) ɞɨɤɚɡɚɧɨ ɞɥɹ ɜɫɟɯ ɧɚɬɭɪɚɥɶɧɵɯ ɯ. ɇɨ ɥɟɜɚɹ ɢ ɩɪɚɜɚɹ ɱɚɫɬɢ
ɪɚɜɟɧɫɬɜɚ (19) ɹɜɥɹɸɬɫɹ ɦɧɨɝɨɱɥɟɧɚɦɢ ɫɬɟɩɟɧɢ n, ɚ ɤɚɤ ɢɡɜɟɫɬɧɨ, ɦɧɨɝɨɱɥɟɧɵ
ɫɬɟɩɟɧɢ n ɪɚɜɧɵ, ɟɫɥɢ ɨɧɢ ɩɪɢɧɢɦɚɸɬ ɨɞɢɧɚɤɨɜɵɟ ɡɧɚɱɟɧɢɹ ɜ (n + 1) ɬɨɱɤɚɯ.
Ɍɟɨɪɟɦɚ ɞɨɤɚɡɚɧɚ.
ɉɨɥɶɡɭɹɫɶ ɬɟɦ, ɱɬɨ ɦɧɨɝɨɱɥɟɧɵ ɪɚɜɧɵ ɜ ɬɨɦ ɢ ɬɨɥɶɤɨ ɜ ɬɨɦ ɫɥɭɱɚɟ, ɟɫɥɢ
ɫɨɜɩɚɞɚɸɬ ɢɯ ɤɨɷɮɮɢɰɢɟɧɬɵ
, ɩɪɢɞɭɦɚɣɬɟ ɞɪɭɝɨɟ ɞɨɤɚɡɚɬɟɥɶɫɬɜɨ ɬɟɨɪɟɦɵ 15,
ɤɚɤ ɷɬɨ ɛɵɥɨ ɫɞɟɥɚɧɨ ɫ ɱɢɫɥɚɦɢ ɋɬɢɪɥɢɧɝɚ 1-ɝɨ ɪɨɞɚ s(n, k).
Ʉɚɤ ɧɚɦ ɭɠɟ ɢɡɜɟɫɬɧɨ, ɱɢɫɥɚ ɋɬɢɪɥɢɧɝɚ 1-ɝɨ ɪɨɞɚ s(n, k) ɨɩɪɟɞɟɥɹɸɬɫɹ
ɤɚɤ ɤɨɷɮɮɢɰɢɟɧɬɵ ɦɧɨɝɨɱɥɟɧɚ [x]
, ɩɨɷɬɨɦɭ ɦɚɬɪɢɰɵ, ɫɨɫɬɚɜɥɟɧɧɵɟ ɢɡ ɱɢɫɟɥ
n
S(n, k) ɢ s(n, k), ɜɡɚɢɦɧɨ ɨɛɪɚɬɧɵɟ.
ɑɢɫɥɨ ɜɫɟɯ ɪɚɡɛɢɟɧɢɣ n-ɷɥɟɦɟɧɬɧɨɝɨ ɦɧɨɠɟɫɬɜɚ ɧɚɡɵɜɚɟɬɫɹ ɱɢɫɥɨɦ
Ȼɟɥɥɚ ɢ ɨɛɨɡɧɚɱɚɟɬɫɹ B
.
n
Ⱦɨɤɚɡɚɬɶ ɫɚɦɨɫɬɨɹɬɟɥɶɧɨ ɫɥɟɞɭɸɳɭɸ ɪɟɤɭɪɪɟɧɬɧɭɸ ɮɨɪɦɭɥɭ, ɫɜɹɡɚɧɧɭɸ ɫ ɱɢɫɥɚɦɢ Ȼɟɥɥɚ:
n
BCB
¦
1
mmnn
m
0
ɉɪɢɫɬɭɩɢɦ ɬɟɩɟɪɶ ɤ ɧɚɩɢɫɚɧɢɸ ɚɥɝɨɪɢɬɦɚ ɝɟɧɟɪɢɪɨɜɚɧɢɹ ɜɫɟɯ ɪɚɡɛɢɟɧɢɣ n-ɷɥɟɦɟɧɬɧɨɝɨ ɦɧɨɠɟɫɬɜɚ. Ɋɚɡɛɢɟɧɢɟ ɦɧɨɠɟɫɬɜɚ {1, 2, …, n} ɛɭɞɟɦ ɩɪɟɞ-
ɫɬɚɜɥɹɬɶ ɫ ɩɨɦɨɳɶɸ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɢ ɛɥɨɤɨɜ, ɭɩɨɪɹɞɨɱɟɧɧɨɣ ɩɨ ɜɨɡɪɚɫɬɚɧɢɸ ɫɚɦɨɝɨ ɦɚɥɟɧɶɤɨɝɨ ɷɥɟɦɟɧɬɚ ɜ ɛɥɨɤɟ. ɗɬɨɬ ɧɚɢɦɟɧɶɲɢɣ ɷɥɟɦɟɧɬ ɛɥɨɤɚ
ɛɭɞɟɦ ɧɚɡɵɜɚɬɶ ɧɨɦɟɪɨɦ ɛɥɨɤɚ. Ɉɬɦɟɬɢɦ, ɱɬɨ ɧɨɦɟɪɚ ɫɨɫɟɞɧɢɯ ɛɥɨɤɨɜ, ɜɨɨɛɳɟ ɝɨɜɨɪɹ, ɧɟ ɹɜɥɹɸɬɫɹ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɵɦɢ ɧɚɬɭɪɚɥɶɧɵɦɢ
ɱɢɫɥɚɦɢ. ȼ ɷɬɨɦ
ɚɥɝɨɪɢɬɦɟ ɦɵ ɛɭɞɟɦ ɢɫɩɨɥɶɡɨɜɚɬɶ ɩɟɪɟɦɟɧɧɵɟ w1[i], w[i], i = 1, …, n, ɫɨɞɟɪ-
64
ɇȿɄɈɌɈɊɕȿ ɋȼȿȾȿɇɂə ɂɁ ȺɅȽȿȻɊɕ ɂ ɄɈɆȻɂɇȺɌɈɊɂɄɂ
ɠɚɳɢɟ, ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ, ɧɨɦɟɪ ɩɪɟɞɵɞɭɳɟɝɨ ɢ ɧɨɦɟɪ ɫɥɟɞɭɸɳɟɝɨ ɛɥɨɤɚ ɞɥɹ
ɛɥɨɤɚ ɫ ɧɨɦɟɪɨɦ i (w[i] = 0, ɟɫɥɢ ɛɥɨɤ ɫ ɧɨɦɟɪɨɦ i ɹɜɥɹɟɬɫɹ ɩɨɫɥɟɞɧɢɦ ɛɥɨɤɨɦ
ɪɚɡɛɢɟɧɢɹ). Ⱦɥɹ ɤɚɠɞɨɝɨ ɷɥɟɦɟɧɬɚ i ɧɨɦɟɪ ɛɥɨɤɚ, ɫɨɞɟɪɠɚɳɟɝɨ ɷɥɟɦɟɧɬ i,
ɛɭɞɟɬ ɯɪɚɧɢɬɶɫɹ ɜ ɩɟɪɟɦɟɧɧɨɣ u[i], ɧɚɩɪɚɜɥɟɧɢɟ, ɜ ɤɨɬɨɪɨɦ «ɞɜɢɠɟɬɫɹ»
ɷɥɟɦɟɧɬ i, ɛɭɞɟɬ ɡɚɤɨɞɢɪɨɜɚɧɨ ɜ ɛɭɥɟɜɫɤɨɣ ɩɟɪɟɦɟɧɧɨɣ v[i] (v[i] = 1, ɟɫɥɢ i
ɞɜɢɠɟɬɫɹ ɜɩɟɪɟɞ, ɩɨɞɨɛɧɨ ɤɚɤ
ɜ ɚɥɝɨɪɢɬɦɟ 21).
Ⱥɥɝɨɪɢɬɦ 22 (ɝɟɧɟɪɢɪɨɜɚɧɢɟ ɜɫɟɯ ɪɚɡɛɢɟɧɢɣ ɦɧɨɠɟɫɬɜɚ).
function r(){s=''; j=1; while(j!=0){s+='{'+j; for(i=j+1; i<=n; i++)if(u[i]==j)s+=','+i;
j=w[j]; s+='}'}x++; return s}
n=+prompt('ȼɜɟɫɬɢ ɱɢɫɥɨ ɷɥɟɦɟɧɬɨɜ ɦɧɨɠɟɫɬɜɚ n', '4'); x=0;
t=''; for(i=0; i<=n; i++){u[i]=1; v[i]=1; w[i]=0} t+=r()+'\n';
j=n; while(j>1){k=u[j];
if(v[j]){if(w[k]==0){w[k]=j; v1[j]=k; w[j]=0}
if(w[k]>j){v1[j]=k; w[j]=w[k]; v1[w[j]]=j; w[k]=j}
u[j]=w[k]}
else{u[j]=v1[k];
if(k==j)if(w[k]==0)w[v1[k]]=0; else{w[v1[k]]=w[k]; v1[w[k]]=v1[k]}}
t+=r()+'\n'; j=n;
while((j>1)*(v[j]*(u[j]==j)+(1-v[j])*(u[j]==1))){v[j]=1-v[j]; j--}}
'ȼɫɟɝɨ ɪɚɡɛɢɟɧɢɣ '+x+'\n'+t
ɉɪɨɬɨɤɨɥ ɜɵɩɨɥɧɟɧɢɹ ɚɥɝɨɪɢɬɦɚ 22 ɞɥɹ n = 4
ȼɫɟɝɨ ɪɚɡɛɢɟɧɢɣ 15
{1,2,3,4}
{1,2,3}{4}
{1,2}{3}{4}
{1,2}{3,4}
{1,2,4}{3}
{1,4}{2}{3}
{1}{2,4}{3}
{1}{2}{3,4}
{1}{2}{3}{4}
{1}{2,3}{4}
{1}{2,3,4}
{1,4}{2,3}
{1,3,4}{2}
{1,3}{2,4}
{1,3}{2}{4}
Ɂɞɟɫɶ ɡɚɨɞɧɨ ɧɚɯɨɞɢɬɫɹ ɱɢɫɥɨ ɜɫɟɯ ɪɚɡɛɢɟɧɢɣ (ɱɢɫɥɨ Ȼɟɥɥɚ) – ɩɟɪɟɦɟɧ-
ɧɚɹ ɯ, ɚ ɮɭɧɤɰɢɹ r() – ɞɥɹ ɡɚɩɢɫɢ ɩɨɞɦɧɨɠɟɫɬɜɚ (
ȼɦɟɫɬɟ ɫ
ɡɚɞɚɱɟɣ ɨ ɪɚɡɛɢɟɧɢɢ ɦɧɨɠɟɫɬɜɚ ɪɚɫɫɦɚɬɪɢɜɚɸɬ ɡɚɞɚɱɭ ɨ ɪɚɡ-
ɛɥɨɤɚ) ɪɚɡɛɢɟɧɢɹ.
ɛɢɟɧɢɢ ɱɢɫɥɚ: ɫɤɨɥɶɤɢɦɢ ɫɩɨɫɨɛɚɦɢ ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ ɱɢɫɥɨ n ɜ ɜɢɞɟ ɫɭɦɦɵ k
ɧɚɬɭɪɚɥɶɧɵɯ ɱɢɫɟɥ.
65
ɉɊɂȼȺɅɈȼ Ⱥ. Ⱥ. ɗɅȿɆȿɇɌɕ ɉɊɈȽɊȺɆɆɂɊɈȼȺɇɂə ɉɊɂ Ɋȿɒȿɇɂɂ ɆȺɌȿɆȺɌɂɑȿɋɄɂɏ ɁȺȾȺɑ
Ʉɚɠɞɭɸ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɶ ɚ1 ɚ2 … ɚk ɬɚɤɭɸ, ɱɬɨ
+ ɚ2 + … + ɚk
n = ɚ
1
ɧɚɡɵɜɚɸɬ ɪɚɡɛɢɟɧɢɟɦ ɱɢɫɥɚ n ɧɚ k ɫɥɚɝɚɟɦɵɯ. ɑɢɫɥɨ ɪɚɡɛɢɟɧɢɣ ɱɢɫɥɚ n ɧɚ k ɫɥɚɝɚɟɦɵɯ ɛɭɞɟɦ ɨɛɨɡɧɚɱɚɬɶ Ɋ(n, k), ɚ ɱɢɫɥɨ ɜɫɟɯ ɪɚɡɛɢɟɧɢɣ ɱɢɫɥɚ n – ɱɟɪɟɡ Ɋ(n),
ɩɪɢɱɟɦ ɫɱɢɬɚɟɬɫɹ, ɱɬɨ Ɋ(0,0) = Ɋ(0) = 1. Ɉɱɟɜɢɞɧɨ, ɱɬɨ
n
¦
k
knPnP
) ,()(
. ɇɚɩɪɢ-
1
ɦɟɪ, 5 = 4 + 1 = 3 + 2 = 3 + 1 + 1 = 2 + 2 + 1 = 2 + 1 + 1 + 1 = 1 + 1 + 1 + 1 + 1,
ɬɨ ɟɫɬɶ Ɋ(5) = 6.
Ɍɟɨɪɟɦɚ 16. ɑɢɫɥɨ ɪɚɡɛɢɟɧɢɣ ɱɢɫɥɚ n ɧɚ k ɫɥɚɝɚɟɦɵɯ ɪɚɜɧɨ ɱɢɫɥɭ ɪɚɡ-
ɛɢɟɧɢɣ ɱɢɫɥɚ n ɫ ɧɚɢɛɨɥɶɲɢɦ ɫɥɚɝɚɟɦɵɦ, ɪɚɜɧɵɦ k.
Ⱦɨɤɚɡɚɬɟɥɶɫɬɜɨ. ɉɪɢɦɟɧɢɦ ɨɱɟɧɶ ɩɪɨɫɬɨɣ ɫɩɨɫɨɛ ɩɪɟɞɫɬɚɜɥɟɧɢɹ ɪɚɡɛɢɟɧɢɹ ɱɢɫɥɚ, ɧɚɡɵɜɚɟɦɵɣ ɞɢɚɝɪɚɦɦɨɣ Ɏɟɪɪɟɪɫɚ. Ⱦɢɚɝɪɚɦɦɚ Ɏɟɪɪɟɪɫɚ ɞɥɹ
ɪɚɡɛɢɟɧɢɹ n = b
ɪɚɡɛɢɟɧɢɹ, ɩɪɢɱɟɦ i-ɹ ɫɬɪɨɤɚ ɟɫɬɶ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɶ b
+ b2 +…+ bk ɫɨɫɬɨɢɬ ɢɡ k ɫɬɪɨɤ, ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɦ ɫɥɚɝɚɟɦɵɦ
1
ɬɨɱɟɤ.
i
Ʉɚɠɞɨɦɭ ɪɚɡɛɢɟɧɢɸ ɱɢɫɥɚ n ɨɞɧɨɡɧɚɱɧɨ ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɫɨɩɪɹɠɟɧɧɨɟ
ɪɚɡɛɢɟɧɢɟ ɷɬɨɝɨ ɱɢɫɥɚ, ɤɨɬɨɪɨɟ ɩɨɥɭɱɚɟɬɫɹ ɬɪɚɧɫɩɨɡɢɰɢɟɣ ɞɢɚɝɪɚɦɦɵ Ɏɟɪ-
ɪɟɪɫɚ (ɩɟɪɟɦɟɧɚ ɪɨɥɹɦɢ ɫɬɪɨɤ ɢ ɫɬɨɥɛɰɨɜ). ɇɚɩɪɢɦɟɪ,
xx
xxxxxxxx
xxxxxxxx
xxxxxxx
xxxxx
x
x
111334416244616
Ɍɪɚɧɫɩɨɡɢɰɢɹ ɞɢɚɝɪɚɦɦɵ Ɏɟɪɪɟɪɫɚ ɨɩɪɟɞɟɥɹɟɬ ɜɡɚɢɦɧɨ ɨɞɧɨɡɧɚɱɧɨɟ
ɫɨɨɬɜɟɬɫɬɜɢɟ ɦɟɠɞɭ ɪɚɡɛɢɟɧɢɹɦɢ ɱɢɫɥɚ
ɫ ɧɚɢɛɨɥɶɲɢɦ ɫɥɚɝɚɟɦɵɦ, ɪɚɜɧɵɦ
n ɧɚ k ɫɥɚɝɚɟɦɵɯ ɢ ɪɚɡɛɢɟɧɢɣ ɱɢɫɥɚ n
k.
Ɍɟɨɪɟɦɚ ɞɨɤɚɡɚɧɚ.
Ɍɟɨɪɟɦɚ 17
ɪɚɜɧɨ ɱɢɫɥɭ ɪɚɡɛɢɟɧɢɣ ɱɢɫɥɚ
. ɑɢɫɥɨ ɪɚɡɛɢɟɧɢɣ ɱɢɫɥɚ n ɧɚ ɩɨɩɚɪɧɨ ɪɚɡɥɢɱɧɵɯ ɫɥɚɝɚɟɦɵɯ
n ɧɚ ɧɟɱɟɬɧɵɟ ɫɥɚɝɚɟɦɵɟ.
Ⱦɨɤɚɡɚɬɟɥɶɫɬɜɨ. ɍɫɬɚɧɨɜɢɦ ɜɡɚɢɦɧɨ ɨɞɧɨɡɧɚɱɧɨɟ ɫɨɨɬɜɟɬɫɬɜɢɟ ɦɟɠɞɭ
ɷɬɢɦɢ ɪɚɡɛɢɟɧɢɹɦɢ. ɉɭɫɬɶ
ɜ ɪɚɡɛɢɟɧɢɢ
r
ɪɚɡ. Ɋɚɡɥɨɠɢɦ ɤɚɠɞɨɟ ri ɩɨ ɫɬɟɩɟɧɹɦ ɞɜɨɣɤɢ:
i
ɇɚɩɪɢɦɟɪ, 13 = 23 + 22 + 1. ɉɪɨɢɡɜɟɞɟɦ ɬɟɩɟɪɶ ɡɚɦɟɧɭ ri ɫɥɚɝɚɟɦɵɯ bi ɧɚ
ɩɨɩɚɪɧɨ ɪɚɡɥɢɱɧɵɟ:
b
, …, bk – ɧɟɱɟɬɧɵɟ ɱɢɫɥɚ ɢ ɩɭɫɬɶ bi ɩɨɹɜɥɹɟɬɫɹ
1
qq
i
ii
21
q
q
21
...22
bbbr
i
ɞɥɹ ɤɚɠɞɨɝɨ i ɢ ɭɩɨɪɹɞɨɱɢɦ ɩɨɥɭ-
i
...).(...22
!! qqr
21
66
ɇȿɄɈɌɈɊɕȿ ɋȼȿȾȿɇɂə ɂɁ ȺɅȽȿȻɊɕ ɂ ɄɈɆȻɂɇȺɌɈɊɂɄɂ
ɱɟɧɧɵɟ ɧɨɜɵɟ ɫɥɚɝɚɟɦɵɟ. ɉɨɥɭɱɢɦ ɪɚɡɛɢɟɧɢɟ ɱɢɫɥɚ n. Ɍɚɤ ɤɚɤ ɤɚɠɞɨɟ ɧɚɬɭɪɚɥɶɧɨɟ ɱɢɫɥɨ ɟɞɢɧɫɬɜɟɧɧɵɦ ɨɛɪɚɡɨɦ ɩɪɟɞɫɬɚɜɢɦɨ ɜ ɜɢɞɟ ɩɪɨɢɡɜɟɞɟɧɢɹ ɫɬɟɩɟɧɢ ɞɜɨɣɤɢ ɢ ɧɟɱɟɬɧɨɝɨ ɱɢɫɥɚ (ɞɨɤɚɠɢɬɟ ɷɬɨ!), ɬɨ ɜ ɩɨɥɭɱɟɧɧɨɦ ɪɚɡɛɢɟɧɢɢ
ɜɫɟ ɫɥɚɝɚɟɦɵɟ ɪɚɡɥɢɱɧɵ. ɇɚɩɪɢɦɟɪ,
28 = 7 + 5 + 5 + 3 + 3 + 3 + 1 + 1 = 7 + 2 · 5 + 2 · 3 + 1 · 3 + 2 · 1 =
Ɉɛɪɚɬɧɨ, ɤɚɠɞɨɟ ɫɥɚɝɚɟɦɨɟ ɪɚɡɛɢɟɧɢɟ ɧɚ ɩɨɩɚɪɧɨ ɪɚɡɥɢɱɧɵɟ ɩɪɟɞɫɬɚɜɢɦ ɜ ɜɢɞɟ:
ɪ2
7 + 10 + 6 + 3+ 2 = 10 + 7 + 6 + 2 + 3.
q
, ɝɞɟ ɪ ɧɟɱɟɬɧɨɟ ɱɢɫɥɨ, ɚ ɷɬɨ 2q ɧɟɱɟɬɧɵɯ ɱɢɫɟɥ ɪ.
Ɍɟɨɪɟɦɚ ɞɨɤɚɡɚɧɚ.
ȼ ɡɚɞɚɱɚɯ ɤɨɦɛɢɧɚɬɨɪɢɤɢ ɧɚ ɩɨɞɫɱɟɬ ɱɢɫɥɚ ɨɛɴɟɤɬɨɜ ɱɚɫɬɨ ɢɫɤɨɦɵɦ
ɪɟɲɟɧɢɟɦ ɹɜɥɹɟɬɫɹ ɧɟɤɨɬɨɪɚɹ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɶ ɱɢɫɟɥ
ɱɢɫɥɨ ɢɫɤɨɦɵɯ ɨɛɴɟɤɬɨɜ «ɪɚɡɦɟɪɧɨɫɬɢ»
ɪɚɡɛɢɟɧɢɣ ɱɢɫɥɚ, ɬɨ ɦɨɠɟɦ ɩɪɢɧɹɬɶ
ɩɨɞɦɧɨɠɟɫɬɜ
ɞɨɜɚɬɟɥɶɧɨɫɬɢ
n-ɷɥɟɦɟɧɬɧɨɝɨ ɦɧɨɠɟɫɬɜɚ, ɬɨ
a
, a1, … ɫɬɚɜɢɦ ɜ ɫɨɨɬɜɟɬɫɬɜɢɟ ɮɨɪɦɚɥɶɧɭɸ ɫɭɦɦɭ:
0
k. ɇɚɩɪɢɦɟɪ, ɟɫɥɢ ɦɵ ɢɳɟɦ ɱɢɫɥɨ
a
= Ɋ(k), ɟɫɥɢ ɢɳɟɦ ɱɢɫɥɨ k-ɷɥɟɦɟɧɬɧɵɯ
k
¦
k
k
k
Ca
ɢ ɬ.ɞ. ȼ ɷɬɨɦ ɫɥɭɱɚɟ ɩɨɫɥɟ-
nk
k
xaxA ,)(
a
, a1, …, ɝɞɟ ak –
0
ɧɚɡɵɜɚɟɦɭɸ
ɪɚɜɧɵ, ɬɨ ɢ ɪɚɜɧɵ ɢɯ ɤɨɷɮɮɢɰɢɟɧɬɵ
ɩɪɨɢɡɜɨɞɹɳɟɣ ɮɭɧɤɰɢɟɣ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɢ a
, a1,… .
0
Ɂɞɟɫɶ ɤɚɤ ɞɥɹ ɦɧɨɝɨɱɥɟɧɨɜ: ɟɫɥɢ ɩɪɨɢɡɜɨɞɹɳɢɟ ɮɭɧɤɰɢɢ
ɚi = b
.
i
Ⱥ(ɯ) ɢ ȼ(ɯ)
ɉɪɨɢɡɜɨɞɹɳɟɣ ɮɭɧɤɰɢɟɣ ɩɨɫɬɨɹɧɧɨɣ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɢ
Ⱥ = {1, 1, 1, …} ɛɭɞɟɬ ɝɟɨɦɟɬɪɢɱɟɫɤɚɹ ɩɪɨɝɪɟɫɫɢɹ. ɇɚɩɨɦɧɢɦ ɢ ɜɵɜɟɞɟɦ ɟɟ
ɫɭɦɦɭ. ɉɭɫɬɶ |
(x) = 1 + x (1 + x + x2 +…) = 1 + xS(x). Ɂɧɚɱɢɬ,
S
x| < 1 ɢ S(x) = 1 + x + x
f
def
¦
k
k
0
32
...1
xxxx
1
1
x
2
+ x3 + …, ɬɨɝɞɚ
1 ||,
x
.
Ⱦɨɤɚɠɢɬɟ, ɱɬɨ ɞɥɹ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɢ ɱɢɫɟɥ
ɜɨɞɹɳɚɹ ɮɭɧɤɰɢɹ
Ⱥ(ɯ) = (1 – 2ɯ)
–1
.
Ⱦɨɤɚɠɢɬɟ, ɱɬɨ ɞɥɹ ɱɢɫɟɥ ɫɨɱɟɬɚɧɢɣ ɩɪɨɢɡɜɨɞɹɳɚɹ ɮɭɧɤɰɢɹ
Ⱥ = {1, 2, 4, 8, …} ɩɪɨɢɡ-
ɋ
(ɯ) ɪɚɜɧɚ
n
f
¦
kk
nn
0nk
.)1()(
xxCxC
Ⱥ = {a
, a
Ⱦɚɥɟɟ, ɡɚɦɟɬɢɦ, ɱɬɨ ɤɚɠɞɨɟ ɩɨɞɦɧɨɠɟɫɬɜɨ ɦɧɨɠɟɫɬɜɚ
} ɨɞɧɨɡɧɚɱɧɨ ɨɩɪɟɞɟɥɹɟɬɫɹ ɭɤɚɡɚɧɢɟɦ ɱɢɫɥɚ ɩɨɹɜɥɟɧɢɣ ɜ ɧɟɦ ɤɚɠɞɨɝɨ ɢɡ
a
n
, …,
ɷɥɟɦɟɧɬɨɜ (ɧɨɥɶ ɪɚɡ ɩɨɹɜɢɥɫɹ, ɬɨ ɟɫɬɶ ɧɟ ɩɨɹɜɢɥɫɹ, ɢ ɨɞɢɧ ɪɚɡ – ɩɨɹɜɢɥɫɹ),
ɬɨ ɦɨɠɧɨ ɫɱɢɬɚɬɶ, ɱɬɨ ɤɚɠɞɵɣ ɦɧɨɠɢɬɟɥɶ (1 +
ɚ
ɷɥɟɦɟɧɬɭ
ɜ ɩɨɞɦɧɨɠɟɫɬɜɟ: ɧɭɥɶ ɪɚɡ (ɫɥɚɝɚɟɦɨɟ
ɦɧɨɠɟɫɬɜɚ Ⱥ ɢ ɭɤɚɡɵɜɚɟɬ ɧɚ ɱɢɫɥɨ ɟɝɨ ɩɨɹɜɥɟɧɢɣ
i
0
ɯ
ɯ) ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɧɟɤɨɬɨɪɨɦɭ
= 1) ɢ ɨɞɢɧ ɪɚɡ (ɫɥɚɝɚɟɦɨɟ ɯ1 = ɯ).
67
ɉɊɂȼȺɅɈȼ Ⱥ. Ⱥ. ɗɅȿɆȿɇɌɕ ɉɊɈȽɊȺɆɆɂɊɈȼȺɇɂə ɉɊɂ Ɋȿɒȿɇɂɂ ɆȺɌȿɆȺɌɂɑȿɋɄɂɏ ɁȺȾȺɑ
Ɍɨ ɟɫɬɶ ɩɨɞɦɧɨɠɟɫɬɜɨ ɨɞɧɨɡɧɚɱɧɨ ɨɩɪɟɞɟɥɹɟɬɫɹ ɜɵɛɨɪɨɦ ɨɞɧɨɝɨ ɢɡ ɫɥɚɝɚɟɦɵɯ ɢɡ ɤɚɠɞɨɝɨ ɦɧɨɠɢɬɟɥɹ ɩɪɨɢɡɜɟɞɟɧɢɹ (1 +
ɯ)(1 + ɯ)… (1 + ɯ). Ɉɩɪɟ-
ɞɟɥɟɧɧɨɟ ɬɚɤɢɦ ɨɛɪɚɡɨɦ ɫɥɚɝɚɟɦɨɟ ɞɚɟɬ ɜɤɥɚɞ ɜ ɤɨɷɮɮɢɰɢɟɧɬ ɩɪɢ
k – ɦɨɳɧɨɫɬɶ ɧɚɲɟɝɨ ɩɨɞɦɧɨɠɟɫɬɜɚ. ɉɟɪɟɧɟɫɟɦ ɧɚɲɢ ɪɚɫɫɭɠɞɟɧɢɹ ɧɚ
ɝɞɟ
ɩɨɞɦɧɨɠɟɫɬɜɚ ɦɧɨɠɟɫɬɜ ɫ ɩɨɜɬɨɪɟɧɢɹɦɢ. Ɋɚɫɫɦɨɬɪɢɦ ɞɥɹ ɩɪɢɦɟɪɚ ɡɚɞɚɱɭ:
ɫ
«ɇɚɣɬɢ ɱɢɫɥɨ
1·
ɚ
, 4·ɚ4), ɞɥɹ ɜɫɟɯ k = 0, 1, …, 10». ȼ ɫɢɥɭ ɧɚɲɢɯ ɪɚɫɫɭɠɞɟɧɢɣ ɩɪɨɢɡɜɨɞɹ-
3
ɳɚɹ ɮɭɧɤɰɢɹ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɢ
f
k
¦
k
k
0
– k-ɷɥɟɦɟɧɬɧɵɯ ɩɨɞɦɧɨɠɟɫɬɜ ɦɧɨɠɟɫɬɜɚ ɏ = (2·ɚ1, 3·ɚ2,
k
ɫ
, ɫ1, … ɪɚɜɧɚ
0
432322
)1)(1)(1)(1(
xxxxxxxxxxxc
12111098765432
xxxxxxxxxxxx
ɨɬɫɸɞɚ, ɧɚɩɪɢɦɟɪ, 3 ɷɥɟɦɟɧɬɧɵɯ ɩɨɞɦɧɨɠɟɫɬɜɚ ɦɧɨɠɟɫɬɜɚ
ɏ ɪɚɜɧɨ 15,
ɚ ɱɢɫɥɨ 5 ɷɥɟɦɟɧɬɧɵɯ – 22.
ɋɨɨɬɜɟɬɫɬɜɭɸɳɢɦ ɩɨɞɛɨɪɨɦ
ɢɡɜɨɥɶɧɵɟ ɨɝɪɚɧɢɱɟɧɢɹ ɧɚ ɱɢɫɥɨ ɜɯɨɠɞɟɧɢɣ ɷɥɟɦɟɧɬɚ
ɦɧɨɠɢɬɟɥɶ ɢɦɟɟɬ ɜɢɞ (1 +
2
ɯ
ȼ ɱɚɫɬɧɨɫɬɢ, ɟɫɥɢ ɧɚ ɱɢɫɥɨ ɜɯɨɠɞɟɧɢɣ ɷɥɟɦɟɧɬɨɜ
i-ɝɨ ɦɧɨɠɢɬɟɥɹ ɦɨɠɧɨ ɧɚɤɥɚɞɵɜɚɬɶ ɩɪɨ-
ɚ
. ɇɚɩɪɢɦɟɪ, ɷɬɨɬ
i
+ ɯ5), ɟɫɥɢ ɚi ɦɨɠɟɬ ɩɨɹɜɥɹɬɶɫɹ 0, 2 ɢɥɢ 5 ɪɚɡ.
ɚ
(i = 1, 2, …, n) ɧɟ ɧɚɤɥɚ-
i
ɞɵɜɚɟɬɫɹ ɧɢɤɚɤɢɯ ɨɝɪɚɧɢɱɟɧɢɣ, ɬɨ ɩɪɨɢɡɜɨɞɹɳɚɹ ɮɭɧɤɰɢɹ ɢɦɟɟɬ ɜɢɞ:
(1 +
2
ɯ + ɯ
+…)(1 + ɯ + ɯ2 + …)…(1 + ɯ + ɯ2 + …) = (1 – x)–1(1 – x)–1…(1 – x)–1 = (1 – x)–n
ɑɢɬɚɬɟɥɸ, ɡɧɚɤɨɦɨɦɭ ɫɨ ɫɬɟɩɟɧɧɵɦɢ ɪɹɞɚɦɢ, ɧɟ ɫɨɫɬɚɜɢɬ ɬɪɭɞɚ ɧɚɣɬɢ
ɤɨɷɮɮɢɰɢɟɧɬɵ ɫɬɟɩɟɧɧɨɝɨ ɪɹɞɚ
f
k
xaxA
ɢ
k
k!, ɬɨ ɟɫɬɶ
0)(
f
¦
0
k
f
¦
k
ɩɪɨɢɡɜɟɞɟɧɢɹɦɢ Ʉɨɲɢ.
ɧɨɣ ɜ ɬɨɱɤɟ 0, ɞɟɥɟɧɧɨɣ ɧɚ
ɮɚɤɬɚ ɨ ɱɢɫɥɟ ɫɨɱɟɬɚɧɢɣ ɫ ɩɨɜɬɨɪɟɧɢɹɦɢ.
ɉɭɫɬɶ
0)(
¦
k
ɧɚɞ ɧɢɦɢ ɦɨɠɧɨ ɞɟɥɚɬɶ ɪɚɡɥɢɱɧɵɟ ɚɪɢɮɦɟɬɢɱɟɫɤɢɟ ɞɟɣɫɬɜɢɹ:
ɋɤɥɚɞɵɜɚɬɶ:
1)
ɍɦɧɨɠɚɬɶ ɧɚ ɱɢɫɥɨ:
2)
ɍɦɧɨɠɚɬɶ ɩɨ ɩɪɚɜɢɥɭ:
3)
ɬɚɤɢɟ ɩɪɨɢɡɜɟɞɟɧɢɹ ɧɚɡɵɜɚɸɬɫɹ
f
n
C
f
¦
k
k
xaxA
DD
k
0
k
0)1(
xcx
. ɫk ɪɚɜɧɹɟɬɫɹ k-ɣ ɩɪɨɢɡɜɨɞ-
¦
k
k
k
. ɂ ɷɬɨ ɟɫɬɶ ɟɳɟ ɨɞɧɨ ɢɡɜɟɫɬɧɨɝɨ
kn
1
k
xbxB
– ɩɪɨɢɡɜɨɞɹɳɢɟ ɮɭɧɤɰɢɢ. Ɍɨɝɞɚ
k
;)()()(
xbaxBxA
kk
k
;)(
f
k
,)()(
xcxBxA
ɝɞɟ
¦
k
0
k
...
k
babababac
¦
ikikkkk
0110
0
i
ȼɨɨɪɭɠɢɜɲɢɫɶ ɷɬɢɦ ɦɨɳɧɵɦ ɚɩɩɚɪɚɬɨɦ, ɜɟɪɧɟɦɫɹ ɬɟɩɟɪɶ ɤ ɡɚɞɚɱɟ ɪɚɡ-
ɛɢɟɧɢɣ ɱɢɫɥɚ
n. ɋɧɚɱɚɥɚ ɡɚɦɟɬɢɦ, ɱɬɨ ɤɚɠɞɨɟ ɪɚɡɛɢɟɧɢɟ n = a
+ a2 +…+ an
1
k
ɯ
,
...00491520222015941
68
ɇȿɄɈɌɈɊɕȿ ɋȼȿȾȿɇɂə ɂɁ ȺɅȽȿȻɊɕ ɂ ɄɈɆȻɂɇȺɌɈɊɂɄɂ
xxx
ɦɨɠɧɨ ɩɪɟɞɫɬɚɜɢɬɶ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɶɸ <Ȝ1, …, Ȝn>, Ȝi – ɱɢɫɥɨ ɫɥɚɝɚɟɦɵɯ,
ɪɚɜɧɵɯ
ɢ, ɧɚɨɛɨɪɨɬ, ɤɚɠɞɚɹ ɬɚɤɚɹ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɶ <
ɥɹɟɬ ɪɚɡɛɢɟɧɢɟ ɱɢɫɥɚ
ɜɵɲɚɸɳɢɟ ɱɢɫɥɨ
i, ɬɨ ɟɫɬɶ Ȝ
= |{j : a
i
Ɉɛɨɡɧɚɱɢɦ ɱɟɪɟɡ
y. Ɍɨɝɞɚ ɫɩɪɚɜɟɞɥɢɜɨ ɫɥɟɞɭɸɳɟɟ ɭɬɜɟɪɠɞɟɧɢɟ.
= i j = 1, 2, …, k}|. Ɉɱɟɜɢɞɧɨ, ɱɬɨ
j
Ȝ
, …, Ȝn> ɨɞɧɨɡɧɚɱɧɨ ɨɩɪɟɞɟ-
1
n.
P
(n) ɱɢɫɥɨ ɪɚɡɛɢɟɧɢɣ ɱɢɫɥɚ n ɧɚ ɫɥɚɝɚɟɦɵɟ, ɧɟ ɩɪɟ-
y
n
ni
O
i
i
1¦
Ɍɟɨɪɟɦɚ 18
Ɋ
(0), Py(1), … ɪɚɜɧɚ
y
(1 +
2
ɯ + ɯ
+ …)(1 + ɯ2 + ɯ4 + …)…(1 + ɯy + ɯ
. ɉɪɨɢɡɜɨɞɹɳɚɹ ɮɭɧɤɰɢɹ ɞɥɹ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɢ
2y
+ …) = (1 – x)
–1
(1 – x2)
–1
…(1 – xy)–1
Ⱦɨɤɚɡɚɬɟɥɶɫɬɜɨ. ɉɪɚɜɚɹ ɱɚɫɬɶ ɪɚɜɟɧɫɬɜɚ ɹɜɥɹɟɬɫɹ ɩɪɨɢɡɜɟɞɟɧɢɟɦ y
ɛɟɫɤɨɧɟɱɧɵɯ ɫɭɦɦ. ɉɨɥɶɡɭɹɫɶ ɩɪɚɜɢɥɨɦ ɩɪɨɢɡɜɟɞɟɧɢɹ ɩɪɨɢɡɜɨɞɹɳɢɯ ɮɭɧɤɰɢɣ, ɪɚɫɤɪɵɜɚɹ ɫɤɨɛɤɢ ɩɨɥɭɱɢɦ ɫɭɦɦɭ ɫɥɚɝɚɟɦɵɯ ɜɢɞɚ
ɧɨɦɟɪ ɱɥɟɧɚ, ɜɵɛɪɚɧɧɨɝɨ ɢɡ
ɰɢɟɧɬ ɩɪɢ
y
O
¦
i
1
i
n
ɯ
ɪɚɜɟɧ ɱɢɫɥɭ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɟɣ ɜɢɞɚ <Ȝ1, …, Ȝy>, ɬɚɤɢɯ, ɱɬɨ
ni
ɢ ɡɧɚɱɢɬ ɪɚɜɟɧ Py(n).
i-ɣ ɛɟɫɤɨɧɟɱɧɨɣ ɫɭɦɦɵ. Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɤɨɷɮɮɢ-
2
OO
21
...
y
O
y
, ɝɞɟ Ȝi –
Ɍɟɨɪɟɦɚ ɞɨɤɚɡɚɧɚ.
Ɍɟɨɪɟɦɚ 19
. ɉɪɨɢɡɜɨɞɹɳɚɹ ɮɭɧɤɰɢɹ ɞɥɹ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɢ
Ɋ(0), P(1), … ɪɚɜɧɚ
k
f
1
12422
kyy
)1(...)...1...)...(1...)(1(
xxxxxxx
Ⱦɨɤɚɡɚɬɟɥɶɫɬɜɨ ɬɟɨɪɟɦɵ ɚɧɚɥɨɝɢɱɧɨ ɩɪɟɞɵɞɭɳɟɦɭ, ɧɨ ɜɵ ɩɪɨɜɟɞɢɬɟ
ɟɝɨ ɫɚɦɨɫɬɨɹɬɟɥɶɧɨ.
Ⱦɥɹ ɩɪɢɦɟɪɚ ɩɪɢɜɟɞɟɦ ɟɳɟ ɨɞɧɨ
ɚ
, ɚ2, …, ɝɞɟ ɚn – ɱɢɫɥɨ ɪɚɡɛɢɟɧɢɣ ɱɢɫɥɚ n ɧɚ ɩɨɩɚɪɧɨ ɪɚɡɥɢɱɧɵɟ ɫɥɚɝɚɟɦɵɟ.
1
ɞɨɤɚɡɚɬɟɥɶɫɬɜɨ ɬɟɨɪɟɦɵ 17. ɉɭɫɬɶ
Ɍɨɝɞɚ ɩɪɨɢɡɜɨɞɹɳɚɹ ɮɭɧɤɰɢɹ ɷɬɨɣ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɢ ɪɚɜɧɚ
Ⱥ(ɯ) = (1 + ɯ)(1 + ɯ
2
) … (1 + ɯk)…
Ⱥ ɩɪɨɢɡɜɨɞɹɳɚɹ ɮɭɧɤɰɢɹ ɞɥɹ ɪɚɡɛɢɟɧɢɣ ɧɚ ɧɟɱɟɬɧɵɟ ɫɥɚɝɚɟɦɵɟ ɛɭɞɟɬ
ȼ(ɯ) = (1 – ɯ)
Ⱦɚɥɟɟ, ɬɚɤ ɤɚɤ (1 +
1
1
x
)(
xA
1
1
x
–1
k
x
) = (1 – x2k)/(1 – xk), ɬɨ
42
1
x
2
1
x
(1 – ɯ3)
6
x
...
–1
…(1 – ɯ
1
1
2k–1) –1
…
1
1
1
533
1
xxxx
)(...
xB
69
,
ɉɊɂȼȺɅɈȼ Ⱥ. Ⱥ. ɗɅȿɆȿɇɌɕ ɉɊɈȽɊȺɆɆɂɊɈȼȺɇɂə ɉɊɂ Ɋȿɒȿɇɂɂ ɆȺɌȿɆȺɌɂɑȿɋɄɂɏ ɁȺȾȺɑ
ɉɪɨɢɡɜɨɞɹɳɢɟ ɮɭɧɤɰɢɢ ɷɬɢɯ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɟɣ ɪɚɜɧɵ, ɡɧɚɱɢɬ,
ɢ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɢ ɪɚɜɧɵ.
Ɍɟɩɟɪɶ ɨɩɢɲɟɦ ɚɥɝɨɪɢɬɦ, ɦɨɞɟɥɢɪɭɸɳɢɣ ɜɫɟ ɪɚɡɛɢɟɧɢɹ ɡɚɞɚɧɧɨɝɨ ɱɢɫ-
ɥɚ
n. Ʉɚɠɞɨɟ ɪɚɡɛɢɟɧɢɟ ɜ ɷɬɨɦ ɚɥɝɨɪɢɬɦɟ ɩɪɟɞɫɬɚɜɥɟɧɨ ɩɟɪɟɦɟɧɧɵɦɢ u[1] >
u[2] > … > u[d] ɢ ɩɟɪɟɦɟɧɧɵɦɢ v[1], v[2], …, v[n], ɝɞɟ v[i] ɫɨɞɟɪɠɢɬ ɢɧɮɨɪɦɚɰɢɸ ɨ ɬɨɦ, ɫɤɨɥɶɤɨ ɪɚɡ ɫɥɚɝɚɟɦɨɟ u[i] ɩɨɹɜɥɹɟɬɫɹ ɜ ɪɚɡɛɢɟɧɢɢ. ɗɬɨ ɩɨɡɜɨɥɹɟɬ
ɧɚɯɨɞɢɬɶ ɫɥɟɞɭɸɳɟɟ ɪɚɡɛɢɟɧɢɟ ɡɚ ɱɢɫɥɨ ɲɚɝɨɜ, ɨɝɪɚɧɢɱɟɧɧɨɣ ɧɟɤɨɬɨɪɨɣ ɩɨɫɬɨɹɧɧɨɣ, ɧɟ ɡɚɜɢɫɹɳɟɣ ɨɬ
n.
Ⱥɥɝɨɪɢɬɦ 23 (ɧɚɯɨɠɞɟɧɢɟ ɜɫɟɯ ɪɚɡɛɢɟɧɢɣ ɱɢɫɥɚ).
function r(){p=''; q=''; for(i=1; i<=d; i++){p+=u[i]+'*'+v[i]+'+';
for(j=0; j<v[i]; j++)q+=u[i]+'+'; }
return q.substr(0,q.length-1)+'='+p.substr(0,p.length-1)}
n=+prompt('ȼɜɟɫɬɢ ɱɢɫɥɨ n','7');
u[1]=n; v[1]=1; d=1; t=r();
while(u[1]>1){s=0; if(u[d]==1){s+=v[d]; d--}
s+=u[d]; v[d]--; l=u[d]-1;
if(v[d]>0)d++;
u[d]=l; v[d]=floor(s/l); l=s%l;
if(l!=0){d++; u[d]=l; v[d]=1}
t+='\n'+r(); }'ȼɫɟɝɨ ɪɚɡɛɢɟɧɢɣ '+t.split('\n').length+'\n'+t
ɉɪɨɬɨɤɨɥ ɜɵɩɨɥɧɟɧɢɹ ɚɥɝɨɪɢɬɦɚ 23 ɞɥɹ n = 7
ȼɫɟɝɨ ɪɚɡɛɢɟɧɢɣ 15
7=7*1
6+1=6*1+1*1
5+2=5*1+2*1
5+1+1=5*1+1*2
4+3=4*1+3*1
4+2+1=4*1+2*1+1*1
4+1+1+1=4*1+1*3
3+3+1=3*2+1*1
3+2+2=3*1+2*2
3+2+1+1=3*1+2*1+1*2
3+1+1+1+1=3*1+1*4
2+2+2+1=2*3+1*1
2+2+1+1+1=2*2+1*3
2+1+1+1+1+1=2*1+1*5
1+1+1+1+1+1+1=1*7
Ɂɞɟɫɶ ɡɚɨɞɧɨ ɧɚɯɨɞɢɬɫɹ ɱɢɫɥɨ ɜɫɟɯ ɪɚɡɛɢɟɧɢɣ ɱɢɫɥɚ
n, ɚ ɮɭɧɤɰɢɹ r() –
ɞɥɹ ɡɚɩɢɫɢ ɨɱɟɪɟɞɧɨɝɨ ɪɚɡɛɢɟɧɢɹ ɢ t – ɫɩɢɫɨɤ ɪɚɡɛɢɟɧɢɣ.
70
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