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ɇȿɄɈɌɈɊɕȿ ɋȼȿȾȿɇɂə ɂɁ ȺɅȽȿȻɊɕ ɂ ɄɈɆȻɂɇȺɌɈɊɂɄɂ
ɧɚ ɛɨɥɶɲɟɦ ɪɚɫɫɬɨɹɧɢɢ ɨɬ Į, ɬɨ ɟɫɬɶ ɞɥɹ ɥɸɛɵɯ ɰɟɥɵɯ ɫ ɢ b d > 0 ɬɚɤɢɯ, ɱɬɨ
c
a
z
ɜɵɩɨɥɧɹɟɬɫɹ ɧɟɪɚɜɟɧɫɬɜɨ:
d
b
c
d
a
!
.
DD
b
ɋɥɟɞɭɸɳɚɹ ɬɟɨɪɟɦɚ ɞɚɟɬ ɨɬɜɟɬ ɧɚ ɩɨɫɬɚɜɥɟɧɧɭɸ ɡɚɞɚɱɭ.
Ɍɟɨɪɟɦɚ 11. ȼɫɹɤɨɟ ɧɚɢɥɭɱɲɟɟ ɩɪɢɛɥɢɠɟɧɢɟ ɱɢɫɥɚ Į ɟɫɬɶ ɨɞɧɚ ɢɡ ɩɨɞɯɨɞɹɳɢɯ ɢɥɢ ɩɪɨɦɟɠɭɬɨɱɧɵɯ ɞɪɨɛɟɣ ɢɡɨɛɪɚɠɚɸɳɟɣ ɷɬɨ ɱɢɫɥɨ ɰɟɩɧɨɣ ɞɪɨɛɢ.
Ʉ ɫɨɠɚɥɟɧɢɸ, ɩɪɢɦɟɧɟɧɢɟ ɷɬɨɣ ɬɟɨɪɟɦɵ ɡɚɬɪɭɞɧɟɧɨ ɨɬɱɚɫɬɢ ɩɨɬɨɦɭ,
ɱɬɨ ɩɨɞɯɨɞɹɳɢɟ ɞɪɨɛɢ ɞɚɸɬ ɜɟɫɶɦɚ ɯɨɪɨɲɟɟ ɩɪɢɛɥɢɠɟɧɢɟ ɱɢɫɥɚ
, ɚ1, …]. Ⱥ ɢɦɟɧɧɨ, ɢɡ ɫɜɨɣɫɬɜ 5 ɢ 4 ɥɟɝɤɨ ɜɵɜɟɫɬɢ ɨɰɟɧɤɭ:
Į = [ɚ
0
p
1
D
)(
qqq
nnn
1
n
d
qqq
11
nnn
.
ɗɬɨ ɜ ɫɨɜɨɤɭɩɧɨɫɬɢ ɫ ɤɨɦɩɶɸɬɟɪɧɨɣ ɨɲɢɛɤɨɣ ɦɨɠɟɬ ɩɪɢɜɟɫɬɢ
ɤ ɨɲɢɛɨɱɧɨɦɭ ɬɨɥɤɨɜɚɧɢɸ ɫɜɨɣɫɬɜ ɱɢɫɥɚ Į. Ʉɪɨɦɟ ɬɨɝɨ, ɧɟɩɨɥɧɵɟ ɱɚɫɬɧɵɟ ɚ
ɱɚɫɬɨ ɛɵɜɚɸɬ ɨɱɟɧɶ ɛɨɥɶɲɢɦɢ, ɱɬɨ ɡɚɦɟɞɥɹɟɬ ɩɨɢɫɤ ɧɭɠɧɨɣ ɩɪɨɦɟɠɭɬɨɱɧɨɣ
ɞɪɨɛɢ. ɂɧɵɦɢ ɫɥɨɜɚɦɢ, ɟɫɬɶ ɧɚɞ ɱɟɦ ɩɨɞɭɦɚɬɶ… Ⱥɥɝɨɪɢɬɦ 19, ɩɪɢɜɟɞɟɧɧɵɣ
ɧɢɠɟ, ɞɥɹ ɧɚɯɨɠɞɟɧɢɹ ɧɭɠɧɨɣ ɩɪɨɦɟɠɭɬɨɱɧɨɣ ɞɪɨɛɢ ɩɪɢɦɟɧɹɟɬ ɛɢɧɚɪɧɵɣ
ɩɨɢɫɤ ɢ ɞɚɟɬ ɩɪɢɟɦɥɟɦɭɸ ɢɧɮɨɪɦɚɰɢɸ ɫ ɬɨɱɧɨɫɬɶɸ ɞɨ eps = 10
–12
. Ɉɞɧɚɤɨ
ɨɩɬɢɦɚɥɶɧɵɦ ɟɝɨ ɧɚɡɜɚɬɶ ɧɟɥɶɡɹ ɢ ɜɚɲɟɣ ɡɚɞɚɱɟɣ ɹɜɥɹɟɬɫɹ ɭɥɭɱɲɟɧɢɟ ɷɬɨɝɨ
ɚɥɝɨɪɢɬɦɚ.
Ⱥɥɝɨɪɢɬɦ 19 (ɩɪɢɛɥɢɠɟɧɢɟ ɱɢɫɥɚ ɨɛɵɤɧɨɜɟɧɧɨɣ ɞɪɨɛɶɸ).
t=prompt('ȼɜɟɫɬɢ ɯ', '2*3-3/7'); x=eval(t); t+='='+x; //eps=1e-12
sg=''; g=''; if(x<0){sg='-'; x=-x; }x0=floor(x); x-=x0;
if(x<eps)g=sg+x0;
else{k=0; k0=0; k1=0; v=''; p=0; q=1; p1=1; q1=0;
p0=1; q0=1; y=x; y0=0; y1=sqrt(eps);
while(1){p0=p1; q0=q1; p1=p; q1=q;
y0=1/y; k=floor(y0); p=p1*k+p0; q=q1*k+q0;
if(abs(p/q-x)<eps)break; y=y0-k;
if(y<y1){k0=1; k1=floor(1/y);
while(k1-k0>1){k= floor((k0+k1)/2);
if(abs((k*p+p1)/(k*q+q1)-x)>eps)k0=k; else k1=k; }
if((k*p+p1)/(k*q+q1)-x<eps){p=k*p+p1; q=k*q+q1; break}}}}
p1=p; q1=q; while(p1){p0=p1; p1=q1%p1; q1=p0}p=p/q1; q=q/q1;
if(g=='')if(q==1)g=sg+(x0+1); else g=sg+x0+'.'+p+'/'+q; t+'='+g.
j
51
ɉɊɂȼȺɅɈȼ Ⱥ. Ⱥ. ɗɅȿɆȿɇɌɕ ɉɊɈȽɊȺɆɆɂɊɈȼȺɇɂə ɉɊɂ Ɋȿɒȿɇɂɂ ɆȺɌȿɆȺɌɂɑȿɋɄɂɏ ɁȺȾȺɑ
Ɂɞɟɫɶ ɜɜɨɞɢɬɫɹ ɱɢɫɥɨ ɯ ɢɥɢ ɟɝɨ ɚɥɝɟɛɪɚɢɱɟɫɤɨɟ ɜɵɪɚɠɟɧɢɟ, ɧɚ ɜɵɯɨɞɟ
ɩɨɥɭɱɚɟɬɫɹ ɩɪɢɛɥɢɠɟɧɢɟ ɯ ɨɛɵɤɧɨɜɟɧɧɨɣ ɞɪɨɛɶɸ. Ɍɨɱɧɨɫɬɶ eps ɦɨɠɧɨ ɦɟɧɹɬɶ, ɜɤɥɸɱɚɹ ɟɟ ɜ ɚɥɝɨɪɢɬɦ. ȼɜɨɞ ɷɥɟɦɟɧɬɚɪɧɵɯ ɮɭɧɤɰɢɣ – ɬɚɤ ɠɟ, ɤɚɤ
ɜ ɚɥɝɨɪɢɬɦɟ 18.
ɉɪɨɬɨɤɨɥ ɜɵɩɨɥɧɟɧɢɹ ɚɥɝɨɪɢɬɦɚ 19 ɞɥɹ x=
Math.sin(Math.PI/6)=0.49999999999999994=0.1/2
ȼ ɡɚɤɥɸɱɟɧɢɟ ɪɚɫɫɦɨɬɪɢɦ ɩɟɪɢɨɞɢɱɟɫɤɢɟ ɰɟɩɧɵɟ ɞɪɨɛɢ.
ɐɟɩɧɚɹ ɞɪɨɛɶ [ɚ
, ɚ1, …] ɧɚɡɵɜɚɟɬɫɹ ɩɟɪɢɨɞɢɱɟɫɤɨɣ, ɟɫɥɢ ɫɭɳɟɫɬɜɭɸɬ
0
ɬɚɤɢɟ ɰɟɥɵɟ ɩɨɥɨɠɢɬɟɥɶɧɵɟ ɱɢɫɥɚ N ɢ p, ɱɬɨ ɞɥɹ ɥɸɛɨɝɨ k N
= ak;
a
k+p
ɚɧɚɥɨɝɢɱɧɨ ɞɟɫɹɬɢɱɧɵɦ ɞɪɨɛɹɦ ɛɭɞɟɦ ɨɛɨɡɧɚɱɚɬɶ ɬɚɤɭɸ ɩɟɪɢɨɞɢɱɟɫɤɭɸ
ɞɪɨɛɶ ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ:
, ɚ1, …, ɚ
[ɚ
0
, (ɚN, …, ɚ
N–1
N+p–1
)].
ɇɚɩɨɦɧɢɦ, ɱɬɨ ɱɢɫɥɨ ɯ ɧɚɡɵɜɚɟɬɫɹ ɤɜɚɞɪɚɬɢɱɟɫɤɨɣ ɢɪɪɚɰɢɨɧɚɥɶɧɨ-
ɫɬɶɸ, ɟɫɥɢ ɨɧɨ ɹɜɥɹɟɬɫɹ ɪɟɲɟɧɢɟɦ ɤɜɚɞɪɚɬɧɨɝɨ ɭɪɚɜɧɟɧɢɹ ɚɯ + bɯ + ɫ = 0
ɫ ɰɟɥɵɦɢ ɤɨɷɮɮɢɰɢɟɧɬɚɦɢ ɚ, b, ɫ.
ɂɡɜɟɫɬɧɚ ɫɥɟɞɭɸɳɚɹ ɬɟɨɪɟɦɚ:
Ɍɟɨɪɟɦɚ 12. ȼɫɹɤɚɹ ɩɟɪɢɨɞɢɱɟɫɤɚɹ ɰɟɩɧɚɹ ɞɪɨɛɶ ɢɡɨɛɪɚɠɚɟɬ ɤɜɚɞɪɚɬɢɱɟɫɤɭɸ ɢɪɪɚɰɢɨɧɚɥɶɧɨɫɬɶ, ɢ, ɨɛɪɚɬɧɨ, ɜɫɹɤɚɹ ɤɜɚɞɪɚɬɢɱɟɫɤɚɹ ɢɪɪɚɰɢɨɧɚɥɶɧɨɫɬɶ ɢɡɨɛɪɚɠɚɟɬɫɹ ɩɟɪɢɨɞɢɱɟɫɤɨɣ ɰɟɩɧɨɣ ɞɪɨɛɶɸ.
Ⱦɨɤɚɡɚɬɟɥɶɫɬɜɨ. ɉɟɪɜɨɟ ɭɬɜɟɪɠɞɟɧɢɟ ɞɨɤɚɡɵɜɚɟɬɫɹ
ɧɟɫɥɨɠɧɨ. ɉɭɫɬɶ
Į = [ɚ
, ɚ1, …, ɚ
0
, (ɚN, …, ɚ
N-1
N+p-1
)],
= r
ɬɨɝɞɚ r
ɢ ɜ ɫɢɥɭ ɫɜɨɣɫɬɜɚ 8 ɢɦɟɟɦ
N
N+p
prp
21
D
NNN
qrq
21
NNN
prp
21
pNNpN
.
qrq
21
pNNpN
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɱɢɫɥɨ r
ɭɞɨɜɥɟɬɜɨɪɹɟɬ ɤɜɚɞɪɚɬɧɨɦɭ ɭɪɚɜɧɟɧɢɸ
N
ɫ ɰɟɥɵɦɢ ɤɨɷɮɮɢɰɢɟɧɬɚɦɢ, ɬɨ ɟɫɬɶ ɹɜɥɹɟɬɫɹ ɤɜɚɞɪɚɬɢɱɟɫɤɨɣ ɢɪɪɚɰɢɨɧɚɥɶɧɨɫɬɶɸ, ɧɨ ɬɨɝɞɚ ɜ ɫɢɥɭ ɫɜɨɣɫɬɜɚ 8 ɢ Į ɟɫɬɶ ɤɜɚɞɪɚɬɢɱɟɫɤɚɹ ɢɪɪɚɰɢɨɧɚɥɶɧɨɫɬɶ.
Ɉɛɪɚɬɧɨɟ ɭɬɜɟɪɠɞɟɧɢɟ ɞɨɤɚɡɵɜɚɟɬɫɹ ɧɟɫɤɨɥɶɤɨ ɫɥɨɠɧɟɟ, ɩɨɷɬɨɦɭ ɩɪɨɜɟɞɢɬɟ ɟɝɨ ɫɚɦɨɫɬɨɹɬɟɥɶɧɨ.
Ɂɚɦɟɬɢɦ, ɱɬɨ ɞɥɹ ɚɥɝɟɛɪɚɢɱɟɫɤɢɯ ɢɪɪɚɰɢɨɧɚɥɶɧɨɫɬɟɣ ɛɨɥɟɟ ɜɵɫɨɤɢɯ
ɫɬɟɩɟɧɟɣ ɧɟɢɡɜɟɫɬɧɨ ɧɢɤɚɤɢɯ ɫɜɨɣɫɬɜ ɢɡɨɛɪɚɠɚɸɳɢɯ ɢɯ ɰɟɩɧɵɯ ɞɪɨɛɟɣ, ɚɧɚ-
52
ɇȿɄɈɌɈɊɕȿ ɋȼȿȾȿɇɂə ɂɁ ȺɅȽȿȻɊɕ ɂ ɄɈɆȻɂɇȺɌɈɊɂɄɂ
o
M
ɥɨɝɢɱɧɵɯ ɬɟɨɪɟɦɟ 6. Ʉɪɨɦɟ ɬɨɝɨ, ɞɨ ɧɚɫɬɨɹɳɟɝɨ ɜɪɟɦɟɧɢ ɧɟɢɡɜɟɫɬɧɨ ɪɚɡɥɨɠɟɧɢɟ ɜ ɰɟɩɧɭɸ ɞɪɨɛɶ ɧɢ ɨɞɧɨɝɨ ɚɥɝɟɛɪɚɢɱɟɫɤɨɝɨ ɱɢɫɥɚ ɫɬɟɩɟɧɢ ɜɵɲɟ ɞɜɭɯ.
ɇɟɢɡɜɟɫɬɧɨ, ɦɨɠɟɬ ɥɢ ɬɚɤɨɟ ɪɚɡɥɨɠɟɧɢɟ ɢɦɟɬɶ ɨɝɪɚɧɢɱɟɧɧɵɟ ɷɥɟɦɟɧɬɵ; ɧɟɢɡɜɟɫɬɧɨ, ɦɨɠɟɬ ɥɢ ɨɧɨ ɢɦɟɬɶ ɧɟɨɝɪɚɧɢɱɟɧɧɵɣ ɪɹɞ ɷɥɟɦɟɧɬɨɜ, ɢ ɬ.ɞ. ɇɚɩɪɢɦɟɪ,
3
ɷɬɨ ɧɟɢɡɜɟɫɬɧɨ ɞɥɹ
.
2
ɇɚ ɡɚɧɹɬɢɹɯ ɜ ɆɉȽɍ ɛɵɥɚ ɩɨɫɬɚɜɥɟɧɚ ɡɚɞɚɱɚ: «Ɉɬɜɟɬɨɦ ɤɨɦɩɶɸɬɟɪɧɨɣ
ɩɪɨɝɪɚɦɦɵ ɹɜɥɹɟɬɫɹ ɱɢɫɥɨ Į, ɩɪɨ ɤɨɬɨɪɨɟ ɢɡɜɟɫɬɧɨ, ɱɬɨ ɨɧɨ ɹɜɥɹɟɬɫɹ ɤɜɚɞɪɚ-
ɬɢɱɟɫɤɨɣ ɢɪɪɚɰɢɨɧɚɥɶɧɨɫɬɶɸ:
2
D
bacb24
, ɝɞɟ a, b, c – ɰɟɥɵɟ ɱɢɫɥɚ.
a
Ɍɪɟɛɭɟɬɫɹ ɧɚɣɬɢ a, b, c». Ɇɵ (ɚɜɬɨɪ ɢ ɟɝɨ ɫɬɭɞɟɧɬɵ) ɧɟ ɫɦɨɝɥɢ ɧɚɩɢɫɚɬɶ ɞɨɫɬɚɬɨɱɧɨ ɩɪɢɟɦɥɟɦɵɣ ɚɥɝɨɪɢɬɦ ɧɟ ɬɨɥɶɤɨ ɞɥɹ ɷɬɨɣ ɡɚɞɚɱɢ, ɧɨ ɢ ɞɚɠɟ ɞɥɹ ɬɚ-
ɤɨɣ: ɩɨ «ɤɨɦɩɶɸɬɟɪɧɨɦɭ»
D
7yx
, ɝɞɟ ɯ ɢ ɭ ɪɚɰɢɨɧɚɥɶɧɵɟ, ɧɚɣɬɢ ɯ ɢ ɭ.
ɉɨɩɵɬɤɚ ɭɜɢɞɟɬɶ ɩɟɪɢɨɞɢɱɧɨɫɬɶ ɫ ɩɨɦɨɳɶɸ ɚɥɝɨɪɢɬɦɚ 18 ɭɫɩɟɯɚ ɧɟ ɢɦɟɥɚ.
1
ɇɚɩɪɢɦɟɪ, ɞɥɹ
7 x
7 x
ɢ
ɩɨɥɭɱɢɥɨɫɶ
5
Math.sqrt(7)=[2,1,1,1,4,1,1,1,4,1,1,1,4,1,1,1,4,1,1,1,4,1,1,1,4,1,2]=2.31869902/493
53213=2.6457513110645907, eps=1e-20
Math.sqrt(7)+1/5=[2,1,5,2,14,4,5,21,1,6,134]=2.70265989/83081147=2.845751311
064591, eps=1e-20
Ⱦɥɹ ɩɟɪɜɨɝɨ ɱɢɫɥɚ ɩɟɪɢɨɞ ɩɪɨɫɥɟɠɢɜɚɟɬɫɹ (1, 1, 1, 4), ɚ ɞɥɹ ɜɬɨɪɨɝɨ – ɧɟɬ.
§ 7. ɉɟɪɟɫɬɚɧɨɜɤɢ
ȼɨɡɜɪɚɳɚɹɫɶ ɤ ɝɪɭɩɩɚɦ, ɩɪɢɜɟɞɟɦ ɨɱɟɧɶ ɜɚɠɧɵɣ ɩɪɢɦɟɪ, ɚ ɢɦɟɧɧɨ –
ɫɢɦɦɟɬɪɢɱɟɫɤɭɸ ɝɪɭɩɩɭ ɧɚ ɦɧɨɠɟɫɬɜɟ.
ɉɭɫɬɶ A – ɩɪɨɢɡɜɨɥɶɧɨɟ ɦɧɨɠɟɫɬɜɨ. Ɋɚɫɫɦɨɬɪɢɦ ɦɧɨɠɟɫɬɜɨ S ɜɫɟɯ
ɫɸɪɴɟɤɬɢɜɧɵɯ ɜɡɚɢɦɧɨ ɨɞɧɨɡɧɚɱɧɵɯ ɮɭɧɤɰɢɣ S = {f: A o A} (ɛɢɟɤɰɢɣ). ȼɜɟɞɟɦ
ɧɚ ɷɬɨɦ
ɦɧɨɠɟɫɬɜɟ ɡɚɤɨɧ ɤɨɦɩɨɡɢɰɢɢ ɤɚɤ ɫɭɩɟɪɩɨɡɢɰɢɸ ɮɭɧɤɰɢɣ f
ɢ gS: fg = f(g). Ɍɨɝɞɚ S ɹɜɥɹɟɬɫɹ ɝɪɭɩɩɨɣ, ɬɚɤ ɤɚɤ ɫɭɩɟɪɩɨɡɢɰɢɹ ɞɜɭɯ ɛɢɟɤɰɢɣ
ɟɫɬɶ ɛɢɟɤɰɢɹ, ɨɩɟɪɚɰɢɹ ɫɭɩɟɪɩɨɡɢɰɢɢ ɚɫɫɨɰɢɚɬɢɜɧɚ, ɟɟ ɧɟɣɬɪɚɥɶɧɵɣ ɷɥɟɦɟɧɬ – ɬɨɠɞɟɫɬɜɟɧɧɨɟ ɨɬɨɛɪɚɠɟɧɢɟ ɟ: A o A – ɟɫɬɶ ɛɢɟɤɰɢɹ, ɞɥɹ ɜɫɹɤɨɣ ɛɢɟɤɰɢɢ f: A o A ɨɬɨɛɪɚɠɟɧɢɟ f
ɟɤɰɢɟɣ ɢ ɜɵɩɨɥɧɟɧɵ ɪɚɜɟɧɫɬɜɚ f
–1
, ɨɛɪɚɬɧɚɹ ɮɭɧɤɰɢɢ f , ɨɩɪɟɞɟɥɟɧɨ, ɹɜɥɹɟɬɫɹ ɛɢ-
–1
–1
f = ff
= ɟ.
ɗɬɭ ɝɪɭɩɩɭ ɧɚɡɵɜɚɸɬ ɫɢɦɦɟɬɪɢɱɟɫɤɨɣ ɝɪɭɩɩɨɣ ɦɧɨɠɟɫɬɜɚ Ⱥ, ɚ ɜ ɬɨɦ
ɫɥɭɱɚɟ, ɤɨɝɞɚ Ⱥ ɤɨɧɟɱɧɨ, – ɝɪɭɩɩɨɣ ɩɨɞɫɬɚɧɨɜɨɤ ɢɥɢ ɝɪɭɩɩɨɣ ɩɟɪɟɫɬɚɧɨɜɨɤ
ɦɧɨɠɟɫɬɜɚ Ⱥ. ȿɫɥɢ Ⱥ ɫɨɫɬɨɢɬ ɢɡ n ɷɥɟɦɟɧɬɨɜ, ɝɪɭɩɩɭ ɩɟɪɟɫɬɚɧɨɜɨɤ ɷɬɨɝɨ
ɦɧɨɠɟɫɬɜɚ ɧɚɡɵɜɚɸɬ ɫɢɦɦɟɬɪɢɱɟɫɤɨɣ ɝɪɭɩɩɨɣ ɫɬɟɩɟɧɢ n ɢɥɢ ɝɪɭɩɩɨɣ ɩɨɞɫɬɚɧɨɜɨɤ n-ɣ ɫɬɟɩɟɧɢ ɢ ɨɛɨɡɧɚɱɚɸɬ S
ɉɭɫɬɶ G ɢ G
– ɝɪɭɩɩɵ. Ɏɭɧɤɰɢɹ
1
.
n
: GG
ɧɚ
ɧɚɡɵɜɚɟɬɫɹ ɝɨɦɨɦɨɪ-
1
ɮɢɡɦɨɦ, ɟɫɥɢ ɞɥɹ ɥɸɛɵɯ ɷɥɟɦɟɧɬɨɜ ɯ ɢ ɭ ɝɪɭɩɩɵ G
53
ɉɊɂȼȺɅɈȼ Ⱥ. Ⱥ. ɗɅȿɆȿɇɌɕ ɉɊɈȽɊȺɆɆɂɊɈȼȺɇɂə ɉɊɂ Ɋȿɒȿɇɂɂ ɆȺɌȿɆȺɌɂɑȿɋɄɂɏ ɁȺȾȺɑ
o
ij(ɯɭ) = ij(ɯ)ij(ɭ).
Ⱦɨɤɚɠɟɦ, ɱɬɨ ij(ɟ) = ɟ
ȼ ɫɚɦɨɦ ɞɟɥɟ, ɟɫɥɢ ɯ
1G1
ɢ ij(ɯ–1) = (ij(ɯ))–1, ɝɞɟ ɟ1 – ɟɞɢɧɢɰɚ ɝɪɭɩɩɵ G1.
1
, ɬɨ ɫɭɳɟɫɬɜɭɟɬ ɬɚɤɨɣ ɷɥɟɦɟɧɬ ɯG, ɱɬɨ ij(ɯ) = ɯ1.
Ɉɬɫɸɞɚ
ɯ
= ij(ɯ) = ij(ɯɟ) = ij(ɯ)ij(ɟ) = ɯ1ij(ɟ) = ɯ1ɟ1.
1
Ⱥɧɚɥɨɝɢɱɧɨ ɯ
= ij(ɟ) ɯ1 ɢ, ɫɥɟɞɨɜɚɬɟɥɶɧɨ, ij(ɟ) = ɟ1. Ⱦɚɥɟɟ,
1
= ij(ɟ) = ij(ɯ–1ɯ) = ij(ɯɯ–1) = ij(ɯ)ij(ɯ–1) = ɯ1ij(ɯ–1) = ij(ɯ–1)ɯ1,
ɟ
1
–1
ɨɬɤɭɞɚ (ij(ɯ))
– ɨɛɪɚɬɧɵɣ ɤ ɯ1.
ȿɫɥɢ ɝɨɦɨɦɨɪɮɢɡɦ ij ɹɜɥɹɟɬɫɹ ɜɡɚɢɦɧɨ ɨɞɧɨɡɧɚɱɧɵɦ ɨɬɨɛɪɚɠɟɧɢɟɦ, ɬɨ ij
ɧɚɡɵɜɚɟɬɫɹ ɢɡɨɦɨɪɮɢɡɦɨɦ, ɩɪɢ ɷɬɨɦ ɝɨɜɨɪɹɬ, ɱɬɨ ɝɪɭɩɩɵ G ɢ G
ȿɫɥɢ ɝɪɭɩɩɚ G ɢɡɨɦɨɪɮɧɚ ɧɟɤɨɬɨɪɨɣ ɩɨɞɝɪɭɩɩɟ ɇ G
ɝɪɭɩɩɚ G ɢɡɨɦɨɪɮɧɨ ɜɤɥɚɞɵɜɚɟɬɫɹ ɜ ɝɪɭɩɩɭ G
.
1
ɢɡɨɦɨɪɮɧɵ.
1
, ɬɨ ɝɨɜɨɪɹɬ, ɱɬɨ
1
ȼ ɪɹɞɟ ɫɥɭɱɚɟɜ ɩɪɢ ɢɡɭɱɟɧɢɢ ɝɪɭɩɩ ɛɵɜɚɟɬ ɭɞɨɛɧɨ ɪɚɫɫɦɚɬɪɢɜɚɬɶ ɧɟ
ɫɚɦɢ ɝɪɭɩɩɵ G, ɚ ɝɪɭɩɩɵ ɢɥɢ ɩɨɞɝɪɭɩɩɵ, ɢɡɨɦɨɪɮɧɵɟ G. ɇɚ ɷɬɨɬ ɫɱɟɬ ɢɡɜɟɫɬɧɚ ɫɥɟɞɭɸɳɚɹ ɬɟɨɪɟɦɚ.
Ɍɟɨɪɟɦɚ 13 (ɬɟɨɪɟɦɚ Ʉɷɥɢ). Ʌɸɛɚɹ ɝɪɭɩɩɚ ɢɡɨɦɨɪɮɧɨ ɜɤɥɚɞɵɜɚɟɬɫɹ
ɜ ɫɢɦɦɟɬɪɢɱɟɫɤɭɸ ɝɪɭɩɩɭ ɧɟɤɨɬɨɪɨɝɨ ɦɧɨɠɟɫɬɜɚ Ⱥ.
ɗɬɚ ɬɟɨɪɟɦɚ ɭɤɚɡɵɜɚɟɬ ɧɚ ɬɨɬ ɮɚɤɬ, ɱɬɨ, ɢɡɭɱɚɹ ɤɚɤɭɸ-ɥɢɛɨ ɝɪɭɩɩɭ
, ɞɨɫɬɚɬɨɱɧɨ ɪɚɫɫɦɨɬɪɟɬɶ ɧɟɤɨɬɨɪɭɸ ɩɨɞɝɪɭɩɩɭ ɫɢɦɦɟɬɪɢɱɟɫɤɨɣ ɝɪɭɩɩɵ, ɚ ɤɨɝɞɚ
ɝɪɭɩɩɚ ɤɨɧɟɱɧɚ, ɬɨ ɩɨɞɝɪɭɩɩɭ S
ȼ ɤɚɱɟɫɬɜɟ ɦɧɨɠɟɫɬɜɚ Ⱥ ɫɢɦɦɟɬɪɢɱɟɫɤɨɣ ɝɪɭɩɩɵ S
Ⱥ={1, 2, …, n}. ɗɥɟɦɟɧɬɵ ɷɬɨɣ ɝɪɭɩɩɵ, ɬɨ ɟɫɬɶ ɮɭɧɤɰɢɢ
ɜɚɸɬ ɩɟɪɟɫɬɚɧɨɜɤɚɦɢ ɢ ɨɛɨɡɧɚɱɚɸɬ f = <a
n, ɢɥɢ ɩɨɞɫɬɚɧɨɜɤɚɦɢ
f
ɇɟɣɬɪɚɥɶɧɵɣ ɷɥɟɦɟɧɬ – ɩɨɞɫɬɚɧɨɜɤɭ
– ɝɪɭɩɩɵ ɩɟɪɟɫɬɚɧɨɜɨɤ.
n
1
n
...21
§
¨
¨
21
©
·
¸
¸
aaa
...
n
¹
ɨɛɵɱɧɨ ɜɵɛɢɪɚɸɬ
n
ɧɚ
:
AAf
, ɧɚɡɵ-
, a2, … , an>, ɝɞɟ ai = f(i), i = 1, 2, …,
.
n
...21
§
¨
e
¨
©
·
ɧɚɡɵɜɚɸɬ ɬɨɠ-
¸
¸
n
...21
¹
ɞɟɫɬɜɟɧɧɨɣ ɩɟɪɟɫɬɚɧɨɜɤɨɣ. Ⱦɥɹ ɩɪɢɦɟɪɚ ɪɚɫɫɦɨɬɪɢɦ ɞɜɟ ɩɨɞɫɬɚɧɨɜɤɢ
54321
§
¨
f
¨
©
54321
·
ɢ
¸
¸
41235
¹
§
g
¨
¨
©
·
.
¸
¸
35142
¹
Ɍɨɝɞɚ ɢɯ ɩɪɨɢɡɜɟɞɟɧɢɟɦ ɛɭɞɟɬ ɩɨɞɫɬɚɧɨɜɤɚ h:
§
fgh
¨
¨
©
54321
·
.
¸
¸
24513
¹
54
ɇȿɄɈɌɈɊɕȿ ɋȼȿȾȿɇɂə ɂɁ ȺɅȽȿȻɊɕ ɂ ɄɈɆȻɂɇȺɌɈɊɂɄɂ
Ɂɚɦɟɬɢɦ, ɱɬɨ ɷɬɚ ɨɩɟɪɚɰɢɹ ɧɟɤɨɦɦɭɬɚɬɢɜɧɚ, ɬɨ ɟɫɬɶ fg gf :
54321
§
gffg
¨
¨
©
·
¸
¸
24513
¹
§
z
¨
¨
©
54321
§
·
¨
¸
¨
¸
35142
©
¹
54321
§
·
¨
¸
¨
¸
41235
©
¹
54321
·
.
¸
¸
52413
¹
ȼɨɡɶɦɟɦ ɧɟɤɨɬɨɪɵɣ ɷɥɟɦɟɧɬ, ɧɚɩɪɢɦɟɪ ɚ
, ɬɨɝɞɚ ɩɨɥɭɱɢɦ ɰɟɩɨɱɤɭ
1
o f(ɚ1) o f(f(ɚ1)) o … o ɚ1.
ɚ
1
ɍɛɟɞɢɬɟɫɶ, ɱɬɨ ɬɚɤɢɟ ɰɟɩɨɱɤɢ ɜɫɟɝɞɚ ɢɦɟɸɬ ɦɟɫɬɨ ɞɥɹ ɥɸɛɨɝɨ ɚ
. ɇɚɡɵɜɚɸɬɫɹ
j
ɬɚɤɢɟ ɰɟɩɨɱɤɢ ɰɢɤɥɚɦɢ. Ⱦɚɞɢɦ ɛɨɥɟɟ ɫɬɪɨɝɨɟ ɨɩɪɟɞɟɥɟɧɢɟ.
ɐɢɤɥɨɦ ɞɥɢɧɵ k ɧɚɡɵɜɚɟɬɫɹ ɩɟɪɟɫɬɚɧɨɜɤɚ f ؝ [b
, b2, … bk}{1, 2, …, n} ɬɚɤɚɹ, ɱɬɨ f(b1) = b2, f(b2) = b3, …, f(bk) = b1
{b
1
ɢ f(ɯ) = ɯ ɞɥɹ ɥɸɛɨɝɨ ɯ{1, 2, …, n}/{b
, b2, …, bk}.
1
, b2,… bk], ɝɞɟ
1
Ʌɟɝɤɨ ɩɨɤɚɡɚɬɶ (ɭɛɟɞɢɬɟɫɶ ɜ ɷɬɨɦ!), ɱɬɨ ɥɸɛɭɸ ɩɟɪɟɫɬɚɧɨɜɤɭ ɦɨɠɧɨ
ɩɪɟɞɫɬɚɜɢɬɶ ɜ ɜɢɞɟ ɩɪɨɢɡɜɟɞɟɧɢɹ ɰɢɤɥɨɜ. ɇɚɩɪɢɦɟɪ,
87654321
§
f
¨
¨
©
·
¸
¸
86741235
¹
].7,6][4,5,1][8][3,2[]8][7,6][3,2][4,5,1[
Ɍɚɤɨɟ ɩɪɟɞɫɬɚɜɥɟɧɢɟ ɩɟɪɟɫɬɚɧɨɜɤɢ ɛɭɞɟɦ ɧɚɡɵɜɚɬɶ ɪɚɡɥɨɠɟɧɢɟɦ ɧɚ
ɰɢɤɥɵ. Ƚɨɜɨɪɹɬ, ɱɬɨ ɩɟɪɟɫɬɚɧɨɜɤɚ f ɟɫɬɶ ɩɟɪɟɫɬɚɧɨɜɤɚ ɬɢɩɚ <Ȝ
ɟɫɥɢ ɨɧɚ ɫɨɞɟɪɠɢɬ ɜ ɪɚɡɥɨɠɟɧɢɢ ɧɚ ɰɢɤɥɵ ɜ ɬɨɱɧɨɫɬɢ Ȝ
i, i = 1, 2, … , n. Ɍɢɩ <Ȝ
O
OO
n
21
(ɟɫɥɢ Ȝi = 0, ɬɨ
n
...21
<5,3,2,1,4,7,6,8> ɢɦɟɟɬ ɬɢɩ 1
ɉɚɪɭ <a
> aj. ɑɢɫɥɨ ɢɧɜɟɪɫɢɣ ɩɟɪɟɫɬɚɧɨɜɤɢ f ɨɛɨɡɧɚɱɢɦ ɱɟɪɟɡ I(f) ɢ ɨɩɪɟɞɟɥɢɦ
ɥɢ a
i
, aj>, i<j, ɧɚɡɵɜɚɸɬ ɢɧɜɟɪɫɢɟɣ ɩɟɪɟɫɬɚɧɨɜɤɢ <a1, a2, …, an>, ɟɫ-
i
ɡɧɚɤ ɷɬɨɣ ɩɟɪɟɫɬɚɧɨɜɤɢ ɤɚɤ sgn(f) ؝ (–1)
, Ȝ2, … , Ȝn> ɨɛɵɱɧɨ ɡɚɩɢɫɵɜɚɟɬɫɹ ɫɢɦɜɨɥɢɱɟɫɤɢ
1
O
i
i
ɨɩɭɫɤɚɟɬɫɹ). ɇɚɩɪɢɦɟɪ, ɩɟɪɟɫɬɚɧɨɜɤɚ
12231
.
I(f)
. ɉɟɪɟɫɬɚɧɨɜɤɭ f ɧɚɡɨɜɟɦ ɱɟɬɧɨɣ,
, Ȝ2, … , Ȝn> ,
1
ɰɢɤɥɨɜ ɞɥɢɧɵ
i
ɟɫɥɢ sgn(f) = 1 ɢ ɧɟɱɟɬɧɨɣ, ɟɫɥɢ sgn(f) = –1.
ɇɚɣɞɢɬɟ ɡɧɚɤ ɢ ɱɢɫɥɨ ɢɧɜɟɪɫɢɣ sgn(ɟ), I(ɟ).
ɉɟɪɟɫɬɚɧɨɜɤɭ, ɹɜɥɹɸɳɭɸɫɹ ɰɢɤɥɨɦ ɞɥɢɧɵ 2, ɧɚɡɵɜɚɸɬ ɬɪɚɧɫɩɨɡɢɰɢɟɣ.
ȼɚɠɧɭɸ ɪɨɥɶ ɢɝɪɚɸɬ ɬɪɚɧɫɩɨɡɢɰɢɢ ɫɨɫɟɞɧɢɯ ɷɥɟɦɟɧɬɨɜ, ɬɨ ɟɫɬɶ ɜɢɞɚ ɬɪɚɧɫɩɨɡɢɰɢɢ [i, i + 1]. ɉɪɢɜɟɞɟɦ ɧɟɤɨɬɨɪɵɟ ɫɜɨɣɫɬɜɚ ɩɟɪɟɫɬɚɧɨɜɨɤ.
ɋɜɨɣɫɬɜɨ 1. ɉɪɨɢɡɜɨɥɶɧɭɸ ɩɟɪɟɫɬɚɧɨɜɤɭ fS
ɦɨɠɧɨ ɩɪɟɞɫɬɚɜɢɬɶ
n
ɜ ɜɢɞɟ ɩɪɨɢɡɜɟɞɟɧɢɹ I(f) ɬɪɚɧɫɩɨɡɢɰɢɣ ɫɨɫɟɞɧɢɯ ɷɥɟɦɟɧɬɨɜ.
Ⱦɨɤɚɡɚɬɟɥɶɫɬɜɨ. Ɂɚɦɟɬɢɦ, ɱɬɨ ɟɫɥɢ f = <a
–1
= t ɢ ɩɪɨɢɡɜɟɞɟɧɢɹ ft ɢɦɟɟɬ ɜɢɞ <a1, a2, … , a
t
ɱɢɫɥɨ ɢɧɜɟɪɫɢɣ, ɪɚɫɩɨɥɨɠɟɧɧɵɯ ɩɟɪɟɞ ɷɥɟɦɟɧɬɨɦ i : ri = |{j : j <i
ɱɟɪɟɡ r
i
a
> ai}|. Ɍɟɩɟɪɶ, ɥɟɝɤɨ ɡɚɦɟɬɢɬɶ, ɱɬɨ ɜ <a1, a2, … , an> ɦɵ ɦɨɠɟɦ ɩɟɪɟɫɬɚɜɥɹɬɶ
j
ɷɥɟɦɟɧɬ
ɧɚ ɩɟɪɜɭɸ ɩɨɡɢɰɢɸ, ɩɪɨɢɡɜɟɞɹ r1 ɬɪɚɧɫɩɨɡɢɰɢɣ ɫɨɫɟɞɧɢɯ ɷɥɟ-
1 ra
1
1
ɦɟɧɬɨɜ, ɡɚɬɟɦ ɷɥɟɦɟɧɬ 2 ɩɟɪɟɫɬɚɜɢɬɶ ɧɚ ɜɬɨɪɭɸ ɩɨɡɢɰɢɸ, ɩɪɨɢɡɜɟɞɹ r
ɡɢɰɢɣ ɫɨɫɟɞɧɢɯ ɷɥɟɦɟɧɬɨɜ ɢ ɬ.ɞ. ȼ ɤɨɧɰɟ ɤɨɧɰɨɜ ɩɨɫɥɟ r
, a2, …, an > ɢ t = [i, i + 1], ɬɨ
1
, a
, ai, a
i–1
i+1
, …, an>. Ɉɛɨɡɧɚɱɢɦ
i+2
+ r2 + … + rn = I(f)
1
2
ɬɪɚɧɫɩɨ-
55
ɉɊɂȼȺɅɈȼ Ⱥ. Ⱥ. ɗɅȿɆȿɇɌɕ ɉɊɈȽɊȺɆɆɂɊɈȼȺɇɂə ɉɊɂ Ɋȿɒȿɇɂɂ ɆȺɌȿɆȺɌɂɑȿɋɄɂɏ ɁȺȾȺɑ
ɲɚɝɨɜ ɩɨɥɭɱɢɦ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɶ <1, 2, …, n>. ɗɬɨ ɨɡɧɚɱɚɟɬ, ɱɬɨ ft1t2…
= e, ɝɞɟ t1, t2, …, t
t
I(f)
– ɬɪɚɧɫɩɨɡɢɰɢɢ ɫɨɫɟɞɧɢɯ ɷɥɟɦɟɧɬɨɜ. Ɉɬɫɸɞɚ
I(f)
1
)(1
1
11)(
.......)...(
ttttttf
1)(
fIfIfI
ɋɜɨɣɫɬɜɨ ɞɨɤɚɡɚɧɨ.
ɋɜɨɣɫɬɜɨ 2. Ⱦɥɹ ɩɪɨɢɡɜɨɥɶɧɵɯ ɩɟɪɟɫɬɚɧɨɜɨɤ f, g S
n
sgn(fg) = sgn(f)·sgn(g).
ɋɜɨɣɫɬɜɨ 3. Ʉɚɠɞɚɹ ɬɪɚɧɫɩɨɡɢɰɢɹ ɟɫɬɶ ɧɟɱɟɬɧɚɹ ɩɟɪɟɫɬɚɧɨɜɤɚ ɢ ɡɧɚɤ
ɰɢɤɥɚ ɞɥɢɧɵ k ɪɚɜɟɧ (–1)
k-1
.
ɋɜɨɣɫɬɜɨ 4. ɐɢɤɥ ɧɟ ɦɟɧɹɟɬɫɹ ɩɪɢ ɤɪɭɝɨɜɨɣ ɩɟɪɟɫɬɚɧɨɜɤɟ ɟɝɨ ɱɥɟɧɨɜ,
ɧɚɩɪɢɦɟɪ, [3, 1, 2] = [2, 3, 1] = [1, 2, 3] z [1, 3, 2].
ɋɜɨɣɫɬɜɨ 5. ɉɟɪɟɫɬɚɧɨɜɤɚ ɧɟ ɢɡɦɟɧɢɬɫɹ, ɟɫɥɢ ɜ ɟɝɨ ɪɚɡɥɨɠɟɧɢɢ ɩɨɦɟ-
ɧɹɬɶ ɦɟɫɬɚɦɢ ɞɜɚ ɰɢɤɥɚ.
Ɂɚɞɚɱɚ. ɇɚɣɞɢɬɟ ɩɨɞɫɬɚɧɨɜɤɭ
§
¨
¨
©
10
10
987654321
·
.
¸
¸
798632154
¹
Ɂɧɚɤ ɩɟɪɟɫɬɚɧɨɜɤɢ ɦɨɠɧɨ ɨɩɪɟɞɟɥɢɬɶ ɫ ɩɨɦɨɳɶɸ ɧɟɩɨɫɪɟɞɫɬɜɟɧɧɨɝɨ
ɩɨɞɫɱɟɬɚ ɜɫɟɯ ɢɧɜɟɪɫɢɣ, ɨɞɧɚɤɨ ɬɚɤɨɣ ɚɥɝɨɪɢɬɦ ɜ ɨɛɳɟɦ ɫɥɭɱɚɟ ɬɪɟɛɭɟɬ ɤɨɥɢɱɟɫɬɜɚ ɲɚɝɨɜ ɬɚɤɨɝɨ ɠɟ ɩɨɪɹɞɤɚ, ɱɬɨ ɢ ɱɢɫɥɨ ɢɧɜɟɪɫɢɣ, ɬɨ ɟɫɬɶ ɩɨ ɦɟɧɶɲɟɣ
ɦɟɪɟ ȍ(n
2
), ɧɚɩɪɢɦɟɪ, I(< n, n – 1, … , 1>) =
)1( nn
. Ɉɞɧɚɤɨ ɦɨɠɧɨ ɫɨɡɞɚɬɶ
2
ɚɥɝɨɪɢɬɦ ɫɥɨɠɧɨɫɬɢ O(n).
Ⱥɥɝɨɪɢɬɦ 20 (ɪɚɡɥɨɠɟɧɢɟ ɩɟɪɟɫɬɚɧɨɜɤɢ ɧɚ ɰɢɤɥɵ).
f='2,1,3,4,6,5,7';
s=0; t=''; b=new Array(); b1=new Array(); b2=new Array();
b=f.split(','); n=b.length; for(i=n; i>0; i--){b[i]=b[i-1]; b1[i]=0; b2[i]=1}
for(i=1; i<=n; i++){if(b2[i]){j=b[i]; s1=1; t+='['+i+',';
while(j!=i){t+=j+','; b2[j]=0; j=b[j]; s++; s1++; if(s>n)break; }
t=t.substr(0,t.length-1)+']'; b1[s1]++}}
s=0; s1=0; t1=''; for(i=1; i<=n; i++){
if(b1[i]!=0)t1+=i+'^'+b1[i]+'*'; s1+=b1[i]*(i%2-1); s+=i*b1[i]}
t1='<'+f+'>='+t+',ɬɢɩ:'+t1.substr(0,t1.length-1);
if(s1%2)t1+=',(ɱɟɬɧɚɹ)'; else t1+=',(ɧɟɱɟɬɧɚɹ)';
if (s!=n)t1+='<'+f+'>-ɧɟ ɩɟɪɟɫɬɚɧɨɜɤɚ ɱɢɫɟɥ {1 ...'+n+'}';
t1+='\ns='+s
ɉɪɨɬɨɤɨɥ ɜɵɩɨɥɧɟɧɢɹ ɚɥɝɨɪɢɬɦɚ 20 ɞɥɹ ɩɟɪɟɫɬɚɧɨɜɤɢ f=<2,1,3,4,6,5,7>
<2,1,3,4,6,5,7> = [1,2][3][4][5,6][7],ɬɢɩ:1^3*2^2,(ɧɟɱɟɬɧɚɹ) s=7
56
ɇȿɄɈɌɈɊɕȿ ɋȼȿȾȿɇɂə ɂɁ ȺɅȽȿȻɊɕ ɂ ɄɈɆȻɂɇȺɌɈɊɂɄɂ
ɗɬɨɬ ɚɥɝɨɪɢɬɦ ɩɪɨɫɦɚɬɪɢɜɚɟɬ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨ ɩɨɡɢɰɢɢ 1, 2, …, n ɩɟɪɟɫɬɚɧɨɜɤɢ, ɢ ɤɚɠɞɵɣ ɪɚɡ, ɤɨɝɞɚ ɷɥɟɦɟɧɬ b[i] ɧɟ ɛɵɥ ɩɪɨɚɧɚɥɢɡɢɪɨɜɚɧ (b2[i]=1),
ɜɵɹɜɥɹɟɬɫɹ ɧɨɜɵɣ ɰɢɤɥ, ɤ ɤɨɬɨɪɨɦɭ ɷɬɨɬ ɷɥɟɦɟɧɬ ɩɪɢɧɚɞɥɟɠɢɬ, ɢ ɬ.ɞ. Ɂɚɨɞɧɨ,
ɨɩɪɟɞɟɥɹɟɬɫɹ, ɹɜɥɹɟɬɫɹ ɥɢ ɜɜɟɞɟɧɧɚɹ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɶ f ɩɟɪɟɫɬɚɧɨɜɤɨɣ.
ɋɭɳɟɫɬɜɭɟɬ ɧɟ ɦɚɥɨ ɚɥɝɨɪɢɬɦɨɜ ɝɟɧɟɪɢɪɨɜɚɧɢɹ ɩɟɪɟɫɬɚɧɨɜɨɤ. ɉɨɫɬɪɨɢɦ ɬɚɤɨɣ ɚɥɝɨɪɢɬɦ, ɜ ɤɨɬɨɪɨɦ ɤɚɠɞɚɹ ɫɥɟɞɭɸɳɚɹ ɩɟɪɟɫɬɚɧɨɜɤɚ ɨɛɪɚɡɭɟɬɫɹ ɢɡ
ɩɪɟɞɵɞɭɳɟɣ ɫ ɩɨɦɨɳɶɸ ɨɞɧɨɤɪɚɬɧɨɣ ɬɪɚɧɫɩɨɡɢɰɢɢ ɫɨɫɟɞɧɢɯ ɷɥɟɦɟɧɬɨɜ.
ɉɪɟɞɩɨɥɨɠɢɦ, ɱɬɨ ɭɠɟ ɩɨɫɬɪɨɟɧɚ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɶ ɩɟɪɟɫɬɚɧɨɜɨɤ ɷɥɟɦɟɧɬɨɜ 2, 3, …, n, ɨɛɥɚɞɚɸɳɚɹ ɷɬɢɦ ɫɜɨɣɫɬɜɨɦ, ɧɚɩɪɢɦɟɪ: 2, 3 ɢ 3, 2 ɞɥɹ n = 3.
Ɍɨɝɞɚ ɬɪɟɛɭɟɦɭɸ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɶ ɩɟɪɟɫɬɚɧɨɜɨɤ ɷɥɟɦɟɧɬɨɜ 1, 2, …, n ɩɨɥɭɱɢɦ, ɜɫɬɚɜɥɹɹ ɷɥɟɦɟɧɬ 1 ɜɫɟɦɢ ɜɨɡɦɨɠɧɵɦɢ ɫɩɨɫɨɛɚɦɢ ɜ ɤɚɠɞɭɸ ɩɟɪɟɫɬɚɧɨɜɤɭ ɷɥɟɦɟɧɬɨɜ 2, 3, …, n. ȼ ɧɚɲɟɦ ɩɪɢɦɟɪɟ ɩɨɥɭɱɚɟɦ:
1 2 3
2 1 3
2 3 1
3 2 1
3 1 2
1 3 2
ȼ ɨɛɳɟɦ ɜɢɞɟ ɷɥɟɦɟɧɬ 1 ɩɟɪɟɦɟɳɚɟɬɫɹ ɩɨɩɟɪɟɦɟɧɧɨ ɦɟɠɞɭ ɩɟɪɜɨɣ
ɢ ɩɨɫɥɟɞɧɟɣ ɫɬɪɨɱɤɨɣ (n – 1)! ɪɚɡ.
Ɍɚɤ ɦɨɠɧɨ ɩɨɥɭɱɢɬɶ ɪɟɤɭɪɫɢɜɧɵɣ ɚɥɝɨɪɢɬɦ, ɧɨ ɬɨɝɞɚ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɶ ɩɟɪɟɫɬɚɧɨɜɨɤ ɩɨɫɬɪɨɢɬɫɹ «ɰɟɥɢɤɨɦ» ɢ ɬɨɥɶɤɨ ɩɨɫɥɟ ɨɤɨɧɱɚɧɢɹ ɜɫɟɝɨ
ɩɨɫɬɪɨɟɧɢɹ ɟɟ ɦɨɠɧɨ ɫɱɢɬɵɜɚɬɶ. Ɍɚɤɨɟ ɪɟɲɟɧɢɟ ɩɨɬɪɟɛɨɜɚɥɨ ɛɵ ɨɝɪɨɦɧɨɝɨ
ɨɛɴɟɦɚ ɩɚɦɹɬɢ, ɜɟɞɶ |S
| = n!. ɉɨɷɬɨɦɭ ɦɵ ɛɭɞɟɦ ɫɬɪɨɢɬɶ ɧɟ ɪɟɤɭɪɫɢɜɧɵɣ ɚɥ-
n
ɝɨɪɢɬɦ. ȼ ɧɟɦ ɞɥɹ ɤɚɠɞɨɝɨ 1 i < n, ɩɟɪɟɦɟɧɧɚɹ w[i] = 0, 1 ɜ ɡɚɜɢɫɢɦɨɫɬɢ ɨɬ
ɬɨɝɨ ɩɟɪɟɧɨɫɢɬɫɹ ɥɢ ɷɥɟɦɟɧɬ ɜɩɟɪɟɞ (w[i] = 1) ɢɥɢ ɧɚɡɚɞ (w[i] = 0) , ɩɟɪɟɦɟɧɧɚɹ v[i] ɩɨɤɚɡɵɜɚɟɬ, ɤɚɤɭɸ ɢɡ ɜɨɡɦɨɠɧɵɯ (n – i + 1) ɷɥɟɦɟɧɬ i ɡɚɧɢɦɚɟɬ ɨɬɧɨɫɢɬɟɥɶɧɨ ɷɥɟɦɟɧɬɨɜ i + 1, …, n
ɧɚ ɫɜɨɟɦ ɩɭɬɢ ɜɩɟɪɟɞ ɢɥɢ ɧɚɡɚɞ. ɉɨɡɢɰɢɸ ɷɥɟɦɟɧɬɚ i ɜ ɬɚɛɥɢɰɟ u (ɦɚɫɫɢɜ ɩɟɪɟɫɬɚɧɚɜɥɢɜɚɟɦɵɯ ɷɥɟɦɟɧɬɨɜ) ɨɩɪɟɞɟɥɹɟɦ ɧɚ
ɨɫɧɨɜɚɧɢɢ ɟɝɨ ɩɨɡɢɰɢɢ ɜ ɛɥɨɤɟ, ɫɨɞɟɪɠɚɳɟɦ i, i + 1, …, n, ɚ ɬɚɤɠɟ ɧɚ ɨɫɧɨɜɚɧɢɢ ɱɢɫɥɚ ɷɥɟɦɟɧɬɨɜ ɢɡ 1, 2, …, i – 1, ɤɨɬɨɪɵɟ ɧɚɯɨɞɹɬɫɹ ɫɥɟɜɚ ɨɬ ɷɬɨɝɨ ɛɥɨɤɚ.
ɗɬɨ ɱɢɫɥɨ j ɜɵɱɢɫɥɹɟɬɫɹ ɤɚɤ ɱɢɫɥɨ ɷɥɟɦɟɧɬɨɜ l < i, ɤɨɬɨɪɵɟ
, ɞɜɢɝɚɹɫɶ ɧɚɡɚɞ,
ɞɨɫɬɢɝɥɢ ɛɵ ɫɜɨɟɝɨ ɤɪɚɣɧɟɝɨ ɥɟɜɨɝɨ ɩɨɥɨɠɟɧɢɹ (v[l] = n – l + 1, w[l] = 0). Ʉɚ-
ɠɞɚɹ ɧɨɜɚɹ ɩɟɪɟɫɬɚɧɨɜɤɚ ɨɛɪɚɡɭɟɬɫɹ ɬɪɚɧɫɩɨɡɢɰɢɟɣ ɫɚɦɨɝɨ ɦɟɧɶɲɟɝɨ ɢɡ ɷɥɟɦɟɧɬɨɜ l, ɤɨɬɨɪɵɣ ɧɟ ɧɚɯɨɞɢɬɫɹ ɜ ɝɪɚɧɢɱɧɨɦ ɩɨɥɨɠɟɧɢɢ (v[l] < n – l + 1) c ɟɝɨ
ɥɟɜɵɦ ɢɥɢ ɩɪɚɜɵɦ ɫɨɫɟɞɨɦ. ɂɬɚɤ,
Ⱥɥɝɨɪɢɬɦ 21 (ɝɟɧɟɪɢɪɨɜɚɧɢɟ ɩɟɪɟɫɬɚɧɨɜɨɤ
ɫ ɦɢɧɢɦɚɥɶɧɵɦ ɱɢɫɥɨɦ ɬɪɚɧɫɩɨɡɢɰɢɣ ɫɨɫɟɞɧɢɯ ɷɥɟɦɟɧɬɨɜ).
f=prompt('ȼɟɫɬɢ
ɱɟɪɟɡ ɡɚɩɹɬɭɸ ɱɬɨ ɩɟɪɟɫɬɚɜɥɹɬɶ','ɤ,ɥ,ɦ,ɧ');
f='0,'+f; n=f.split(',').length-1
for(i=0; i<=n; i++){u[i]=i; v[i]=1; w[i]=1}; v[n]=0;
if(f!='')u=f.split(',');
57
ɉɊɂȼȺɅɈȼ Ⱥ. Ⱥ. ɗɅȿɆȿɇɌɕ ɉɊɈȽɊȺɆɆɂɊɈȼȺɇɂə ɉɊɂ Ɋȿɒȿɇɂɂ ɆȺɌȿɆȺɌɂɑȿɋɄɂɏ ɁȺȾȺɑ
f=u[1]; for(j=2; j<=n; j++)f+=','+u[j]; f+='\n'
i=1; while(i<n){i=1; j=0;
while(v[i]==n-i+1){w[i]=1-w[i]; v[i]=1;
if(w[i])j++; i++}
if(i<n){if(w[i])k=v[i]+j; else k=n-i+1-v[i]+j;
s=u[k]; u[k]=u[k+1]; u[k+1]=s;
f+=u[1]; for(j=2; j<=n; j++)f+=','+u[j]; f+='\n';
v[i]++}}(f.split('\n').length-1)+'\n'+f
Ɋɟɡɭɥɶɬɚɬɨɦ ɪɚɛɨɬɵ ɚɥɝɨɪɢɬɦɚ ɹɜɥɹɟɬɫɹ ɱɢɫɥɨ ɩɨɥɭɱɟɧɧɵɯ ɩɟɪɟɫɬɚɧɨ-
ɜɨɤ. ɂɯ ɫɩɢɫɨɤ:
ɉɪɨɬɨɤɨɥ ɜɵɩɨɥɧɟɧɢɹ ɚɥɝɨɪɢɬɦɚ 21 ɞɥɹ f = <ɤ,ɥ,ɦ,ɧ>
24
ɤ,ɥ,ɦ,ɧ
ɥ,ɤ,ɦ,ɧ
ɥ,ɦ,ɤ,ɧ
ɥ,ɦ,ɧ,ɤ
ɦ,ɥ,ɧ,ɤ
ɦ,ɥ,ɤ,ɧ
ɦ,ɤ,ɥ,ɧ
ɤ,ɦ,ɥ,ɧ
ɤ,ɦ,ɧ,ɥ
ɦ,ɤ,ɧ,ɥ
ɦ,ɧ,ɤ,ɥ
ɦ,ɧ,ɥ,ɤ
ɧ,ɦ,ɥ,ɤ
ɧ,ɦ,ɤ,ɥ
ɧ,ɤ,ɦ,ɥ
ɤ,ɧ,ɦ,ɥ
ɤ,ɧ,ɥ,ɦ
ɧ,ɤ,ɥ,ɦ
ɧ,ɥ,ɤ,ɦ
ɦ,ɤ
ɧ,ɥ,
ɥ,ɧ,ɦ,ɤ
ɥ,ɧ,ɤ,ɦ
ɥ,ɤ,ɧ,ɦ
ɤ,ɥ,ɧ,ɦ
Ɍɟɩɟɪɶ ɩɪɢɫɬɭɩɢɦ ɤ ɪɟɲɟɧɢɸ ɫɮɨɪɦɭɥɢɪɨɜɚɧɧɨɣ ɜɵɲɟ ɡɚɞɚɱɢ
ɨ ɫɥɨɠɧɨɫɬɢ ɚɥɝɨɪɢɬɦɚ 6:
m=a[1]; for(i=2; i<=n; i++) if(a[i]<m)m=a[i]; return m.
58
ɇȿɄɈɌɈɊɕȿ ɋȼȿȾȿɇɂə ɂɁ ȺɅȽȿȻɊɕ ɂ ɄɈɆȻɂɇȺɌɈɊɂɄɂ
Ɂɞɟɫɶ ɬɪɟɛɭɟɬɫɹ ɨɰɟɧɢɬɶ ɱɢɫɥɨ ɨɩɟɪɚɰɢɣ ɩɪɢɫɜɚɢɜɚɧɢɹ «=». Ɍɨ ɟɫɬɶ ɞɥɹ
ɥɸɛɨɝɨ n ɢ k n ɨɩɪɟɞɟɥɢɬɶ ɱɢɫɥɨ ɩɟɪɟɫɬɚɧɨɜɨɤ f
ג
Sn, ɩɪɢ ɨɛɪɚɛɨɬɤɟ ɤɨɬɨɪɵɯ
ɱɢɫɥɨ ɨɩɟɪɚɰɢɣ ɩɪɢɫɜɚɢɜɚɧɢɹ ɜɫɬɪɟɬɢɬɫɹ ɪɨɜɧɨ k ɪɚɡ.
Ⱦɥɹ ɪɟɲɟɧɢɹ ɷɬɨɣ ɡɚɞɚɱɢ ɡɚɩɢɲɟɦ ɷɥɟɦɟɧɬɵ ɩɪɨɢɡɜɨɥɶɧɨɣ ɩɟɪɟɫɬɚɧɨɜ-
ג
Sn, ɜ ɫɬɪɨɤɭ, ɩɨɞɱɟɪɤɢɜɚɹ ɬɟ ɷɥɟɦɟɧɬɵ, ɧɚ ɤɨɬɨɪɵɯ ɩɪɨɢɫɯɨɞɢɬ ɨɩɟɪɚɰɢɹ
ɤɢ f
ɩɪɢɫɜɚɢɜɚɧɢɹ:
. ..., , ..., , , ..., , , ..., , ,
1121 nnnnnn
2211
aaaaaaaa
k
ȼ ɷɬɨɣ ɩɟɪɟɫɬɚɧɨɜɤɟ (k + 1) ɩɪɢɫɜɚɢɜɚɧɢɟ ɪɚɡɛɢɜɚɟɬ ɩɟɪɟɫɬɚɧɨɜɤɭ ɧɚ
(k + 1) ɛɥɨɤ:
111121
12111
aaaaaaaaaa
kkk
ɉɪɢ ɷɬɨɦ ɨɱɟɜɢɞɧɨ, ɱɬɨ ɦɢɧɢɦɚɥɶɧɵɣ ɷɥɟɦɟɧɬ ɜ ɤɚɠɞɨɦ ɛɥɨɤɟ ɫɬɨɢɬ ɧɚ
ɩɟɪɜɨɦ ɦɟɫɬɟ. Ʉɚɠɞɨɣ ɬɚɤɨɣ ɩɟɪɟɫɬɚɧɨɜɤɟ ɩɨɫɬɚɜɢɦ ɜɨ ɜɡɚɢɦɧɨ ɨɞɧɨɡɧɚɱɧɨɟ
ɫɨɨɬɜɟɬɫɬɜɢɟ ɩɟɪɟɫɬɚɧɨɜɤɭ g, ɢɦɟɸɳɟɟ ɫɥɟɞɭɸɳɟɟ ɪɚɡɥɨɠɟɧɢɟ ɧɚ ɰɢɤɥɵ:
]. ..., ,][ ..., ,[...] ..., , ,][ ..., , ,[
12111
111121
aaaaaaaaaag
nnnnnnnn
kkk
Ⱦɥɹ ɩɪɢɦɟɪɚ ɧɚɩɢɲɟɦ ɷɬɨ ɫɨɨɬɜɟɬɫɬɜɢɟ ɞɥɹ n = 3:
ɉɟɪɟɫɬɚɧɨɜɤɚ Ɋɚɡɥɨɠɟɧɢɟ
ɧɚ ɰɢɤɥɵ
ɑɢɫɥɨ
ɰɢɤɥɨɜ
ɑɢɫɥɨ
ɨɩɟɪɚɰɢɣ
ɋɨɨɬɜɟɬɫɬɜɢɟ
f o g
«=»
1 2 3 [1][2][3] 3 1 [1, 2, 3]
1 3 2 [1][2, 3] 2 1 [1, 3, 2]
2 1 3 [1, 2][3] 2 2 [2][1, 3]
2 3 1 [1, 2, 3] 1 2 [2, 3][1]
3 1 2 [1, 3, 2] 1 2 [3][1, 2]
3 2 1 [1, 3][2] 2 3 [3][2][1]
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɦɵ ɩɪɢɯɨɞɢɦ ɤ ɫɥɟɞɭɸɳɟɣ ɡɚɞɚɱɟ. ɇɚɣɬɢ ɱɢɫɥɨ ɩɟɪɟɫɬɚɧɨɜɨɤ f
ג
Sn, ɢɦɟɸɳɢɯ ɜ ɫɜɨɟɦ ɪɚɡɥɨɠɟɧɢɢ ɪɨɜɧɨ k ɰɢɤɥɨɜ.
ɇɚɣɞɟɦ ɞɥɹ ɷɬɢɯ ɱɢɫɟɥ ɪɟɤɭɪɪɟɧɬɧɭɸ ɮɨɪɦɭɥɭ. Ɉɛɨɡɧɚɱɢɦ ɱɟɪɟɡ
Į
– ɱɢɫɥɨ ɩɟɪɟɫɬɚɧɨɜɨɤ ɢɡ Sn, ɢɦɟɸɳɢɟ ɜ ɫɜɨɟɦ ɪɚɡɥɨɠɟɧɢɢ ɪɨɜɧɨ
n,k
k ɰɢɤɥɨɜ. Ⱦɚɥɟɟ, ɷɬɢ ɩɟɪɟɫɬɚɧɨɜɤɢ ɪɚɡɨɛɶɟɦ ɧɚ ɞɜɚ ɧɟɩɟɪɟɫɟɤɚɸɳɢɯɫɹ
ɤɥɚɫɫɚ: ɩɟɪɜɵɣ ɤɥɚɫɫ – ɩɟɪɟɫɬɚɧɨɜɤɢ ɢɡ S
, ɢɦɟɸɳɢɟ ɜ ɫɜɨɟɦ ɪɚɡɥɨɠɟɧɢɢ
n–1
ɪɨɜɧɨ (k – 1) ɰɢɤɥɨɜ, ɭɦɧɨɠɟɧɧɵɟ ɧɚ ɰɢɤɥ [n] ɞɥɢɧɵ 1; ɜɬɨɪɨɣ ɤɥɚɫɫ –
ɰɢɤɥɵ ɩɟɪɟɫɬɚɧɨɜɨɤ ɢɡ S
ɫ ɞɨɛɚɜɥɟɧɧɵɦ ɜ ɧɢɯ ɷɥɟɦɟɧɬɨɦ n. ɉɟɪɜɵɣ ɤɥɚɫɫ ɛɭɞɟɬ ɫɨɞɟɪɠɚɬɶ Į
, ɢɦɟɸɳɢɟ ɜ ɫɜɨɟɦ ɪɚɡɥɨɠɟɧɢɢ ɪɨɜɧɨ k ɰɢɤɥɨɜ
n–1
n–1,k–1
). ..., ,(), ..., ,( ..., ), ..., , ,(), ..., , ,(
nnnnnnnn
59
ɉɊɂȼȺɅɈȼ Ⱥ. Ⱥ. ɗɅȿɆȿɇɌɕ ɉɊɈȽɊȺɆɆɂɊɈȼȺɇɂə ɉɊɂ Ɋȿɒȿɇɂɂ ɆȺɌȿɆȺɌɂɑȿɋɄɂɏ ɁȺȾȺɑ
ɷɥɟɦɟɧɬɨɜ, ɚ ɜɬɨɪɨɣ – (n – 1)Į
ɷɥɟɦɟɧɬɨɜ, ɬɚɤ ɤɚɤ ɟɫɥɢ ɰɢɤɥ
n–1,k
ɜ ɪɚɡɥɨɠɟɧɢɢ ɩɟɪɟɫɬɚɧɨɜɤɢ ɫɨɫɬɨɹɥ ɢɡ m ɷɥɟɦɟɧɬɨɜ, ɬɨ ɷɥɟɦɟɧɬ n ɦɨɠɧɨ
ɞɨɛɚɜɢɬɶ ɜ ɧɟɝɨ m ɫɩɨɫɨɛɚɦɢ, ɩɨɫɬɚɜɢɜ ɟɝɨ ɜɬɨɪɵɦ ɢɥɢ ɬɪɟɬɶɢɦ … ɢɥɢ
ɩɨɫɥɟɞɧɢɦ ɷɥɟɦɟɧɬɨɦ. Ⱥ ɬɚɤ ɤɚɤ ɫɭɦɦɚ ɜɫɟɯ m – ɞɥɢɧ ɰɢɤɥɨɜ, ɪɚɜɧɚ
(n – 1), ɬɨ ɱɢɫɥɨ ɫɩɨɫɨɛɨɜ ɞɨɛɚɜɥɟɧɢɹ ɷɥɟɦɟɧɬɚ n ɜ ɰɢɤɥɵ ɪɚɜɧɨ (n – 1).
Ɉɬɫɸɞɚ
ɩɨɥɭɱɢɦ ɪɟɤɭɪɪɟɧɬɧɭɸ ɮɨɪɦɭɥɭ:
= Į
Į
n,k
n–1,k–1
+ (n – 1)Į
(14)
n–1,k
Ɉɱɟɜɢɞɧɨ, ɱɬɨ ɩɪɢ n 1, Į
= 0, Į
n,0
= 1, ɢ ɦɨɠɧɨ ɫɱɢɬɚɬɶ, ɱɬɨ Į
n,n
0,0
= 1.
ɉɨ ɫɭɬɢ, ɷɬɨ ɟɫɬɶ ɚɛɫɨɥɸɬɧɵɟ ɜɟɥɢɱɢɧɵ ɱɢɫɟɥ ɋɬɢɪɥɢɧɝɚ ɩɟɪɜɨɝɨ ɪɨɞɚ
s(n, k). ɗɬɢ ɱɢɫɥɚ ɬɚɤɠɟ ɧɨɫɹɬ ɧɚɡɜɚɧɢɟ ɱɢɫɟɥ ɋɬɢɪɥɢɧɝɚ. ɑɢɫɥɚ ɋɬɢɪɥɢɧɝɚ
ɩɟɪɜɨɝɨ ɪɨɞɚ – ɷɬɨ ɤɨɷɮɮɢɰɢɟɧɬɵ ɦɧɨɝɨɱɥɟɧɚ
k
؝ ɯ(ɯ – 1)…( ɯ – n + 1) =
[x]
n
¦
nk
xkns0),(
.
Ⱦɨɤɚɠɟɦ ɷɬɨ. Ɉɱɟɜɢɞɧɨ, ɱɬɨ n 1 s(n, n) = 1, s(n, 0) = 0 ɢ s(0, 0) = 1, ɬɚɤ
ɤɚɤ ɩɨ ɨɩɪɟɞɟɥɟɧɢɸ [x]
n
k
n
1
k
0
n
1
¦
k
1
= 1. Ⱦɚɥɟɟ ɞɥɹ n 1 ɢɦɟɟɦ
0
k
n
1
k
1
¦¦
k
0
n
1
¦¦
k
00
k
) ,1()1() ,1(
xknsnxkns
k
n
k
) ,1()1()1)(2)...(1() ,(
xknsnxnxnxxxxkns
).0 ,1()1()1 ,1() ,1()1()1 ,1((
nsnxnnsxknsnkns
Ɇɧɨɝɨɱɥɟɧ ɜ ɥɟɜɨɣ ɱɚɫɬɢ ɩɨɥɭɱɟɧɧɨɝɨ ɪɚɜɟɧɫɬɜɚ ɫɨɜɩɚɥ ɫ ɦɧɨɝɨɱɥɟɧɨɦ
ɩɪɚɜɨɣ, ɡɧɚɱɢɬ, ɪɚɜɧɵ ɢɯ ɤɨɷɮɮɢɰɢɟɧɬɵ, ɢ ɦɵ ɩɨɥɭɱɚɟɦ ɪɟɤɭɪɪɟɧɬɧɭɸ ɮɨɪɦɭɥɭ ɞɥɹ ɱɢɫɟɥ ɋɬɢɪɥɢɧɝɚ ɩɟɪɜɨɝɨ ɪɨɞɚ:
s(n, k) = s(n – 1, k – 1) – (n – 1)s(n –1, k) ɞɥɹ 0 < k < n,
s(n, n) = 1 ɞɥɹ n 0,
s(n, 0) = 0 ɞɥɹ n > 0.
Ɉɬɫɸɞɚ
ɢ ɮɨɪɦɭɥɚ ȼɢɟɬɚ, ɪɟɤɭɪɪɟɧɬɧɵɟ ɮɨɪɦɭɥɵ ɞɥɹ ɤɨɷɮɮɢɰɢɟɧɬɨɜ
ɦɧɨɝɨɱɥɟɧɚ
n
؝ ɯ(ɯ +1)…(ɯ + n – 1) =
[x]
¦
nk
k
xkns0|),(|
(15)
ɢɦɟɸɬ ɜɢɞ:
|s(n,k)| = |s(n –1, k – 1)| + (n – 1)|s(n – 1, k)| ɞɥɹ 0 < k < n,
|s(n,n)| = 1 ɞɥɹ n 0,
|s(n,0)| = 0 ɞɥɹ n > 0.
60
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