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이 항목의 스프링노트 원문주소==
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| 1번째 줄: | 1번째 줄: | ||
| − | <h5 style="margin: 0px; line-height: 3.428em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">이 항목의 스프링노트 원문주소 | + | <h5 style="margin: 0px; line-height: 3.428em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">이 항목의 스프링노트 원문주소== |
* [[q-이항계수의 목록]] | * [[q-이항계수의 목록]] | ||
| 7번째 줄: | 7번째 줄: | ||
| − | <h5 style="margin: 0px; line-height: 3.428em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">개요 | + | <h5 style="margin: 0px; line-height: 3.428em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">개요== |
* 정의<br><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><br> | * 정의<br><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><br> | ||
| 15번째 줄: | 15번째 줄: | ||
| − | <h5 style="margin: 0px; line-height: 2em;">목록 | + | <h5 style="margin: 0px; line-height: 2em;">목록== |
* 다음 목록은 자연수 <math>n</math> 과 <math>r=0,1,\cdots,n</math>에 대한 q-이항계수<br><math>n=1, \{{1,1}\}</math><br><math>n=2, \{{1,1+q,1}\}</math><br><math>n=3, \{{1,1+q+q^2,1+q+q^2,1}\}</math><br><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><br><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><br> 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}<br> 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}<br> 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}<br> 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}<br> 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}<br> | * 다음 목록은 자연수 <math>n</math> 과 <math>r=0,1,\cdots,n</math>에 대한 q-이항계수<br><math>n=1, \{{1,1}\}</math><br><math>n=2, \{{1,1+q,1}\}</math><br><math>n=3, \{{1,1+q+q^2,1+q+q^2,1}\}</math><br><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><br><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><br> 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}<br> 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}<br> 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}<br> 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}<br> 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}<br> | ||
| 23번째 줄: | 23번째 줄: | ||
| − | ==역사 | + | ==역사== |
* [[수학사연표 (역사)|수학사연표]] | * [[수학사연표 (역사)|수학사연표]] | ||
| 31번째 줄: | 31번째 줄: | ||
| − | ==메모 | + | ==메모== |
2012년 11월 1일 (목) 10:21 판
이 항목의 스프링노트 원문주소==
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}
역사
메모
- 정의
\({n \choose r}_q={{[n]_q!} \over {[r]_q![n - r]_q!}}=\frac{(q;q)_n}{(q;q)_r(q;q)_{n-r}}=\frac{(1-q)_q^n}{(1-q)_q^r (1-q)_q^{n-r}}\)
- 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}