"Q-이항계수의 목록"의 두 판 사이의 차이
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Pythagoras0 (토론 | 기여) |
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| 3번째 줄: | 3번째 줄: | ||
* 정의:<math>{n \choose r}_q={{[n]_q!} \over {[r]_q![n - r]_q!}}=\frac{(q;q)_n}{(q;q)_r(q;q)_{n-r}}=\frac{(1-q)_q^n}{(1-q)_q^r (1-q)_q^{n-r}}</math> | * 정의:<math>{n \choose r}_q={{[n]_q!} \over {[r]_q![n - r]_q!}}=\frac{(q;q)_n}{(q;q)_r(q;q)_{n-r}}=\frac{(1-q)_q^n}{(1-q)_q^r (1-q)_q^{n-r}}</math> | ||
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==목록== | ==목록== | ||
| 11번째 줄: | 11번째 줄: | ||
* 다음 목록은 자연수 <math>n</math> 과 <math>r=0,1,\cdots,n</math>에 대한 q-이항계수:<math>n=1, \{{1,1}\}</math>:<math>n=2, \{{1,1+q,1}\}</math>:<math>n=3, \{{1,1+q+q^2,1+q+q^2,1}\}</math>:<math>n=4, \{{1,1+q+q^2+q^3,(1+q^2) (1+q+q^2),1+q+q^2+q^3,1}\}</math>:<math>n=5, \{{1,1+q+q^2+q^3+q^4,(1+q^2) (1+q+q^2+q^3+q^4),(1+q^2) (1+q+q^2+q^3+q^4),1+q+q^2+q^3+q^4,1}\}</math> n=6,{1,1+q+q^2+q^3+q^4+q^5,(1+q^2+q^4) (1+q+q^2+q^3+q^4),(1+q^2) (1+q^3) (1+q+q^2+q^3+q^4),(1+q^2+q^4) (1+q+q^2+q^3+q^4),1+q+q^2+q^3+q^4+q^5,1} n=7,{1,1+q+q^2+q^3+q^4+q^5+q^6,(1+q^2+q^4) (1+q+q^2+q^3+q^4+q^5+q^6),(1-q+q^2) (1+q+q^2+q^3+q^4) (1+q+q^2+q^3+q^4+q^5+q^6),(1-q+q^2) (1+q+q^2+q^3+q^4) (1+q+q^2+q^3+q^4+q^5+q^6),(1+q^2+q^4) (1+q+q^2+q^3+q^4+q^5+q^6),1+q+q^2+q^3+q^4+q^5+q^6,1} n=8,{1,1+q+q^2+q^3+q^4+q^5+q^6+q^7,(1+q^2+q^4+q^6) (1+q+q^2+q^3+q^4+q^5+q^6),(1-q+q^2) (1+q+q^2+q^3+q^4+q^5+q^6) (1+q+q^2+q^3+q^4+q^5+q^6+q^7),(1-q+q^2) (1+q^4) (1+q+q^2+q^3+q^4) (1+q+q^2+q^3+q^4+q^5+q^6),(1-q+q^2) (1+q+q^2+q^3+q^4+q^5+q^6) (1+q+q^2+q^3+q^4+q^5+q^6+q^7),(1+q^2+q^4+q^6) (1+q+q^2+q^3+q^4+q^5+q^6),1+q+q^2+q^3+q^4+q^5+q^6+q^7,1} n=9,{1,1+q+q^2+q^3+q^4+q^5+q^6+q^7+q^8,(1+q^2+q^4+q^6) (1+q+q^2+q^3+q^4+q^5+q^6+q^7+q^8),(1+q^3+q^6) (1+q^2+q^4+q^6) (1+q+q^2+q^3+q^4+q^5+q^6),(1-q+q^2) (1+q^4) (1+q+q^2+q^3+q^4+q^5+q^6) (1+q+q^2+q^3+q^4+q^5+q^6+q^7+q^8),(1-q+q^2) (1+q^4) (1+q+q^2+q^3+q^4+q^5+q^6) (1+q+q^2+q^3+q^4+q^5+q^6+q^7+q^8),(1+q^3+q^6) (1+q^2+q^4+q^6) (1+q+q^2+q^3+q^4+q^5+q^6),(1+q^2+q^4+q^6) (1+q+q^2+q^3+q^4+q^5+q^6+q^7+q^8),1+q+q^2+q^3+q^4+q^5+q^6+q^7+q^8,1} n=10,{1,1+q+q^2+q^3+q^4+q^5+q^6+q^7+q^8+q^9,(1+q^2+q^4+q^6+q^8) (1+q+q^2+q^3+q^4+q^5+q^6+q^7+q^8),(1+q^3+q^6) (1+q^2+q^4+q^6) (1+q+q^2+q^3+q^4+q^5+q^6+q^7+q^8+q^9),(1+q^4) (1+q^3+q^6) (1+q+q^2+q^3+q^4+q^5+q^6) (1+q^2+q^4+q^6+q^8),(1+q) (1-q+q^2) (1+q+q^2) (1+q^4) (1-q+q^2-q^3+q^4) (1+q^3+q^6) (1+q+q^2+q^3+q^4+q^5+q^6),(1+q^4) (1+q^3+q^6) (1+q+q^2+q^3+q^4+q^5+q^6) (1+q^2+q^4+q^6+q^8),(1+q^3+q^6) (1+q^2+q^4+q^6) (1+q+q^2+q^3+q^4+q^5+q^6+q^7+q^8+q^9),(1+q^2+q^4+q^6+q^8) (1+q+q^2+q^3+q^4+q^5+q^6+q^7+q^8),1+q+q^2+q^3+q^4+q^5+q^6+q^7+q^8+q^9,1} | * 다음 목록은 자연수 <math>n</math> 과 <math>r=0,1,\cdots,n</math>에 대한 q-이항계수:<math>n=1, \{{1,1}\}</math>:<math>n=2, \{{1,1+q,1}\}</math>:<math>n=3, \{{1,1+q+q^2,1+q+q^2,1}\}</math>:<math>n=4, \{{1,1+q+q^2+q^3,(1+q^2) (1+q+q^2),1+q+q^2+q^3,1}\}</math>:<math>n=5, \{{1,1+q+q^2+q^3+q^4,(1+q^2) (1+q+q^2+q^3+q^4),(1+q^2) (1+q+q^2+q^3+q^4),1+q+q^2+q^3+q^4,1}\}</math> n=6,{1,1+q+q^2+q^3+q^4+q^5,(1+q^2+q^4) (1+q+q^2+q^3+q^4),(1+q^2) (1+q^3) (1+q+q^2+q^3+q^4),(1+q^2+q^4) (1+q+q^2+q^3+q^4),1+q+q^2+q^3+q^4+q^5,1} n=7,{1,1+q+q^2+q^3+q^4+q^5+q^6,(1+q^2+q^4) (1+q+q^2+q^3+q^4+q^5+q^6),(1-q+q^2) (1+q+q^2+q^3+q^4) (1+q+q^2+q^3+q^4+q^5+q^6),(1-q+q^2) (1+q+q^2+q^3+q^4) (1+q+q^2+q^3+q^4+q^5+q^6),(1+q^2+q^4) (1+q+q^2+q^3+q^4+q^5+q^6),1+q+q^2+q^3+q^4+q^5+q^6,1} n=8,{1,1+q+q^2+q^3+q^4+q^5+q^6+q^7,(1+q^2+q^4+q^6) (1+q+q^2+q^3+q^4+q^5+q^6),(1-q+q^2) (1+q+q^2+q^3+q^4+q^5+q^6) (1+q+q^2+q^3+q^4+q^5+q^6+q^7),(1-q+q^2) (1+q^4) (1+q+q^2+q^3+q^4) (1+q+q^2+q^3+q^4+q^5+q^6),(1-q+q^2) (1+q+q^2+q^3+q^4+q^5+q^6) (1+q+q^2+q^3+q^4+q^5+q^6+q^7),(1+q^2+q^4+q^6) (1+q+q^2+q^3+q^4+q^5+q^6),1+q+q^2+q^3+q^4+q^5+q^6+q^7,1} n=9,{1,1+q+q^2+q^3+q^4+q^5+q^6+q^7+q^8,(1+q^2+q^4+q^6) (1+q+q^2+q^3+q^4+q^5+q^6+q^7+q^8),(1+q^3+q^6) (1+q^2+q^4+q^6) (1+q+q^2+q^3+q^4+q^5+q^6),(1-q+q^2) (1+q^4) (1+q+q^2+q^3+q^4+q^5+q^6) (1+q+q^2+q^3+q^4+q^5+q^6+q^7+q^8),(1-q+q^2) (1+q^4) (1+q+q^2+q^3+q^4+q^5+q^6) (1+q+q^2+q^3+q^4+q^5+q^6+q^7+q^8),(1+q^3+q^6) (1+q^2+q^4+q^6) (1+q+q^2+q^3+q^4+q^5+q^6),(1+q^2+q^4+q^6) (1+q+q^2+q^3+q^4+q^5+q^6+q^7+q^8),1+q+q^2+q^3+q^4+q^5+q^6+q^7+q^8,1} n=10,{1,1+q+q^2+q^3+q^4+q^5+q^6+q^7+q^8+q^9,(1+q^2+q^4+q^6+q^8) (1+q+q^2+q^3+q^4+q^5+q^6+q^7+q^8),(1+q^3+q^6) (1+q^2+q^4+q^6) (1+q+q^2+q^3+q^4+q^5+q^6+q^7+q^8+q^9),(1+q^4) (1+q^3+q^6) (1+q+q^2+q^3+q^4+q^5+q^6) (1+q^2+q^4+q^6+q^8),(1+q) (1-q+q^2) (1+q+q^2) (1+q^4) (1-q+q^2-q^3+q^4) (1+q^3+q^6) (1+q+q^2+q^3+q^4+q^5+q^6),(1+q^4) (1+q^3+q^6) (1+q+q^2+q^3+q^4+q^5+q^6) (1+q^2+q^4+q^6+q^8),(1+q^3+q^6) (1+q^2+q^4+q^6) (1+q+q^2+q^3+q^4+q^5+q^6+q^7+q^8+q^9),(1+q^2+q^4+q^6+q^8) (1+q+q^2+q^3+q^4+q^5+q^6+q^7+q^8),1+q+q^2+q^3+q^4+q^5+q^6+q^7+q^8+q^9,1} | ||
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==역사== | ==역사== | ||
| 19번째 줄: | 19번째 줄: | ||
* [[수학사 연표]] | * [[수학사 연표]] | ||
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==메모== | ==메모== | ||
[[분류:q-급수]] | [[분류:q-급수]] | ||
2020년 12월 28일 (월) 02:58 기준 최신판
개요
- 정의\[{n \choose r}_q={{[n]_q!} \over {[r]_q![n - r]_q!}}=\frac{(q;q)_n}{(q;q)_r(q;q)_{n-r}}=\frac{(1-q)_q^n}{(1-q)_q^r (1-q)_q^{n-r}}\]
목록
- 다음 목록은 자연수 \(n\) 과 \(r=0,1,\cdots,n\)에 대한 q-이항계수\[n=1, \{{1,1}\}\]\[n=2, \{{1,1+q,1}\}\]\[n=3, \{{1,1+q+q^2,1+q+q^2,1}\}\]\[n=4, \{{1,1+q+q^2+q^3,(1+q^2) (1+q+q^2),1+q+q^2+q^3,1}\}\]\[n=5, \{{1,1+q+q^2+q^3+q^4,(1+q^2) (1+q+q^2+q^3+q^4),(1+q^2) (1+q+q^2+q^3+q^4),1+q+q^2+q^3+q^4,1}\}\] n=6,{1,1+q+q^2+q^3+q^4+q^5,(1+q^2+q^4) (1+q+q^2+q^3+q^4),(1+q^2) (1+q^3) (1+q+q^2+q^3+q^4),(1+q^2+q^4) (1+q+q^2+q^3+q^4),1+q+q^2+q^3+q^4+q^5,1} n=7,{1,1+q+q^2+q^3+q^4+q^5+q^6,(1+q^2+q^4) (1+q+q^2+q^3+q^4+q^5+q^6),(1-q+q^2) (1+q+q^2+q^3+q^4) (1+q+q^2+q^3+q^4+q^5+q^6),(1-q+q^2) (1+q+q^2+q^3+q^4) (1+q+q^2+q^3+q^4+q^5+q^6),(1+q^2+q^4) (1+q+q^2+q^3+q^4+q^5+q^6),1+q+q^2+q^3+q^4+q^5+q^6,1} n=8,{1,1+q+q^2+q^3+q^4+q^5+q^6+q^7,(1+q^2+q^4+q^6) (1+q+q^2+q^3+q^4+q^5+q^6),(1-q+q^2) (1+q+q^2+q^3+q^4+q^5+q^6) (1+q+q^2+q^3+q^4+q^5+q^6+q^7),(1-q+q^2) (1+q^4) (1+q+q^2+q^3+q^4) (1+q+q^2+q^3+q^4+q^5+q^6),(1-q+q^2) (1+q+q^2+q^3+q^4+q^5+q^6) (1+q+q^2+q^3+q^4+q^5+q^6+q^7),(1+q^2+q^4+q^6) (1+q+q^2+q^3+q^4+q^5+q^6),1+q+q^2+q^3+q^4+q^5+q^6+q^7,1} n=9,{1,1+q+q^2+q^3+q^4+q^5+q^6+q^7+q^8,(1+q^2+q^4+q^6) (1+q+q^2+q^3+q^4+q^5+q^6+q^7+q^8),(1+q^3+q^6) (1+q^2+q^4+q^6) (1+q+q^2+q^3+q^4+q^5+q^6),(1-q+q^2) (1+q^4) (1+q+q^2+q^3+q^4+q^5+q^6) (1+q+q^2+q^3+q^4+q^5+q^6+q^7+q^8),(1-q+q^2) (1+q^4) (1+q+q^2+q^3+q^4+q^5+q^6) (1+q+q^2+q^3+q^4+q^5+q^6+q^7+q^8),(1+q^3+q^6) (1+q^2+q^4+q^6) (1+q+q^2+q^3+q^4+q^5+q^6),(1+q^2+q^4+q^6) (1+q+q^2+q^3+q^4+q^5+q^6+q^7+q^8),1+q+q^2+q^3+q^4+q^5+q^6+q^7+q^8,1} n=10,{1,1+q+q^2+q^3+q^4+q^5+q^6+q^7+q^8+q^9,(1+q^2+q^4+q^6+q^8) (1+q+q^2+q^3+q^4+q^5+q^6+q^7+q^8),(1+q^3+q^6) (1+q^2+q^4+q^6) (1+q+q^2+q^3+q^4+q^5+q^6+q^7+q^8+q^9),(1+q^4) (1+q^3+q^6) (1+q+q^2+q^3+q^4+q^5+q^6) (1+q^2+q^4+q^6+q^8),(1+q) (1-q+q^2) (1+q+q^2) (1+q^4) (1-q+q^2-q^3+q^4) (1+q^3+q^6) (1+q+q^2+q^3+q^4+q^5+q^6),(1+q^4) (1+q^3+q^6) (1+q+q^2+q^3+q^4+q^5+q^6) (1+q^2+q^4+q^6+q^8),(1+q^3+q^6) (1+q^2+q^4+q^6) (1+q+q^2+q^3+q^4+q^5+q^6+q^7+q^8+q^9),(1+q^2+q^4+q^6+q^8) (1+q+q^2+q^3+q^4+q^5+q^6+q^7+q^8),1+q+q^2+q^3+q^4+q^5+q^6+q^7+q^8+q^9,1}
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