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START Zlajos 17 jun 2007
Extension: If all character once : example: ABCDE......
fixed point: character numbers: | free or "0" | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
"0" | 1 | ||||||||||||
1 | 0 | 1 | |||||||||||
11 | 1 | 0 | 1 | ||||||||||
111 | 2 | 3 | 0 | 1 | |||||||||
1111 | 9 | 8 | 6 | 0 | 1 | ||||||||
11111 | 44 | 45 | 20 | 10 | 0 | 1 | |||||||
111111 | 265 | 264 | 135 | 40 | 15 | 0 | 1 | ||||||
1111111 | 1854 | 1855 | 924 | 315 | 70 | 21 | 0 | 1 |
COMMENT: Analogous to A008290. - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 10 2005
1, 0, 0, 1, 1, 0, 4, 0, 1, 10, 24, 27, 16, 12, 0, 1, 297, 672, 736, 480, 246, 64, 24, 0, 1, 13756, 30480, 32365, 21760, 10300, 3568, 970, 160, 40, 0, 1, 925705, 2016480, 2116836, 1418720, 677655, 243360, 67920, 14688, 2655, 320, 60, 0, 1
fixed point: character numbers: | free or "0" | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
"0" | 1 | ||||||||||||
2 | 0 | 0 | 1 | ||||||||||
22 | 1 | 0 | 4 | 0 | 1 | ||||||||
222 | 10 | 24 | 27 | 16 | 12 | 0 | 1 | ||||||
2222 | 297 | 672 | 736 | 480 | 246 | 64 | 24 | 0 | 1 | ||||
22222 | 13756 | 30480 | 32365 | 21760 | 10300 | 3568 | 970 | 160 | 40 | 0 | 1 | ||
222222 | 925705 | 2016480 | 2116836 | 1418720 | 677655 | 243360 | 67920 | 14688 | 2655 | 320 | 60 | 0 | 1 |
2222222 | 85394646 | 183749160 | 191384599 | 128058000 | 61585776 | 22558928 | 6506955 | 1507392 | 284550 | 43848 | 5901 | 560 | 84 |
If original or classic table: (1.table)
then:
FORMULA: MAPLE p := (x, k)->k!^2*sum(x^j/((k-j)!^2*j!), j=0..k); R := (x, n, k)->p(x, k)^n; f := (t, n, k)->sum(coeff(R(x, n, k), x, j)*(t-1)^j*(n*k-j)!, j=0..n*k);seq(f(0, n, 2)/2!^n, n=0..18); (AUTHOR Barbara Haas Margolius (margolius(AT)math.csuohio.edu) )
fixed point: character numbers: | free or "0" | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
"0" | 1 | ||||||||||||
3 | 0 | 0 | 0 | 1 | |||||||||
33 | 1 | 0 | 9 | 0 | 9 | 0 | 1 | ||||||
333 | 56 | 216 | 378 | 435 | 324 | 189 | 54 | 27 | 0 | 1 | |||
3333 | 13833 | 49464 | 84510 | 90944 | 69039 | 38448 | 16476 | 5184 | 1431 | 216 | 54 | 0 | 1 |
33333 | 6699824 | 23123880 | 38358540 | 40563765 | 30573900 | 17399178 | 7723640 | 2729295 | 776520 | 180100 | 33372 | 5355 | 540 |
333333 | 5691917785 | 19180338840 | 31234760055 | 32659846104 | 24571261710 | 14125889160 | 6433608330 | 2375679240 | 722303568 | 182701480 | 38712600 | 6889320 | 1035330 |
3333333 | 7785547001784 | 25791442770240 | etc |
If original or classic table: (1.table)
then:
FORMULA: MAPLE p := (x, k)->k!^2*sum(x^j/((k-j)!^2*j!), j=0..k); R := (x, n, k)->p(x, k)^n; f := (t, n, k)->sum(coeff(R(x, n, k), x, j)*(t-1)^j*(n*k-j)!, j=0..n*k); seq(f(0, n, 3)/3!^n, n=0..18); (AUTHOR Barbara Haas Margolius (margolius(AT)math.csuohio.edu) [ [4]]
" " :1
111 :2
222 :10
333 :56
444 :346
555 :2252
etc... A000172 Franel number a(n) = Sum C(n,k)^3, k=0..n. [ [5]]
111 :3
222 :24
333 :216
444 :1824
555 :15150
etc... A000279 Card matching. [ [6]] COMMENT
Number of permutations of 3 distinct letters (ABC) each with n copies such that one (1) fixed points. E.g. if AAAAABBBBBCCCCC n=3*5 letters permutations then one fixed points n5=15150 - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Feb 02 2006
111 :0
222 :27
333 :378
444 :4536
555 :48600
etc... A000535 Card matching. [ [7]]
111 :1
222 :16
333 :435
444 :7136
555 :99350
etc... A000489 Card matching. [ [8]]
Zlajos 19. jun. 2007.
Zlajos 28. jun. 2007. 16. apr. 2009.
To show that for n ≥ 1, the expected number of fixed points is 1 :
We'll number the permutations p = 1 to n!
Now let X[p,m]=1 if in the p_th permutation, element m is fixed,
when it is not fixed, X[p,m]=0
Now the expected number of fixed points
is E_n[F] = sum_p_from_1_to_n! { sum_m_from_1_to_n { X[p,m] } } / n!
=> E_n[F] = sum_m_from_1_to_n { sum_p_from_1_to_n! { X[p,m] } } / n!
=> E_n[F] = sum_m_from_1_to_n { (n-1)! } / n!
=> E_n[F] = n * (n-1)! / n!
=> E_n[F] = 1
(with thanks to FD) Pnelnik ( talk) 08:42, 19 July 2009 (UTC)
On the article page the first table is not very pretty. There is no horizontal bar under number 2,3,4,5,6,7
and the verticle bar goes down too far:
0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | |
---|---|---|---|---|---|---|---|---|
0 | 1 | |||||||
1 | 0 | 1 |
Perhaps one solution would be to put in blanks in those extra cells.
0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | |
---|---|---|---|---|---|---|---|---|
0 | 1 | |||||||
1 | 0 | 1 |
It is not ideal, but I think it looks a bit better.
Pnelnik (
talk) 23:23, 19 July 2009 (UTC)
- This is the coefficient operator. Definition can be found at http://en.wikipedia.org/wiki/Formal_power_series#Extracting_coefficients Heycarnut ( talk) 08:37, 10 August 2013 (UTC)
This article is rated Start-class on Wikipedia's
content assessment scale. It is of interest to the following WikiProjects: | |||||||||||||||||||||
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START Zlajos 17 jun 2007
Extension: If all character once : example: ABCDE......
fixed point: character numbers: | free or "0" | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
"0" | 1 | ||||||||||||
1 | 0 | 1 | |||||||||||
11 | 1 | 0 | 1 | ||||||||||
111 | 2 | 3 | 0 | 1 | |||||||||
1111 | 9 | 8 | 6 | 0 | 1 | ||||||||
11111 | 44 | 45 | 20 | 10 | 0 | 1 | |||||||
111111 | 265 | 264 | 135 | 40 | 15 | 0 | 1 | ||||||
1111111 | 1854 | 1855 | 924 | 315 | 70 | 21 | 0 | 1 |
COMMENT: Analogous to A008290. - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 10 2005
1, 0, 0, 1, 1, 0, 4, 0, 1, 10, 24, 27, 16, 12, 0, 1, 297, 672, 736, 480, 246, 64, 24, 0, 1, 13756, 30480, 32365, 21760, 10300, 3568, 970, 160, 40, 0, 1, 925705, 2016480, 2116836, 1418720, 677655, 243360, 67920, 14688, 2655, 320, 60, 0, 1
fixed point: character numbers: | free or "0" | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
"0" | 1 | ||||||||||||
2 | 0 | 0 | 1 | ||||||||||
22 | 1 | 0 | 4 | 0 | 1 | ||||||||
222 | 10 | 24 | 27 | 16 | 12 | 0 | 1 | ||||||
2222 | 297 | 672 | 736 | 480 | 246 | 64 | 24 | 0 | 1 | ||||
22222 | 13756 | 30480 | 32365 | 21760 | 10300 | 3568 | 970 | 160 | 40 | 0 | 1 | ||
222222 | 925705 | 2016480 | 2116836 | 1418720 | 677655 | 243360 | 67920 | 14688 | 2655 | 320 | 60 | 0 | 1 |
2222222 | 85394646 | 183749160 | 191384599 | 128058000 | 61585776 | 22558928 | 6506955 | 1507392 | 284550 | 43848 | 5901 | 560 | 84 |
If original or classic table: (1.table)
then:
FORMULA: MAPLE p := (x, k)->k!^2*sum(x^j/((k-j)!^2*j!), j=0..k); R := (x, n, k)->p(x, k)^n; f := (t, n, k)->sum(coeff(R(x, n, k), x, j)*(t-1)^j*(n*k-j)!, j=0..n*k);seq(f(0, n, 2)/2!^n, n=0..18); (AUTHOR Barbara Haas Margolius (margolius(AT)math.csuohio.edu) )
fixed point: character numbers: | free or "0" | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
"0" | 1 | ||||||||||||
3 | 0 | 0 | 0 | 1 | |||||||||
33 | 1 | 0 | 9 | 0 | 9 | 0 | 1 | ||||||
333 | 56 | 216 | 378 | 435 | 324 | 189 | 54 | 27 | 0 | 1 | |||
3333 | 13833 | 49464 | 84510 | 90944 | 69039 | 38448 | 16476 | 5184 | 1431 | 216 | 54 | 0 | 1 |
33333 | 6699824 | 23123880 | 38358540 | 40563765 | 30573900 | 17399178 | 7723640 | 2729295 | 776520 | 180100 | 33372 | 5355 | 540 |
333333 | 5691917785 | 19180338840 | 31234760055 | 32659846104 | 24571261710 | 14125889160 | 6433608330 | 2375679240 | 722303568 | 182701480 | 38712600 | 6889320 | 1035330 |
3333333 | 7785547001784 | 25791442770240 | etc |
If original or classic table: (1.table)
then:
FORMULA: MAPLE p := (x, k)->k!^2*sum(x^j/((k-j)!^2*j!), j=0..k); R := (x, n, k)->p(x, k)^n; f := (t, n, k)->sum(coeff(R(x, n, k), x, j)*(t-1)^j*(n*k-j)!, j=0..n*k); seq(f(0, n, 3)/3!^n, n=0..18); (AUTHOR Barbara Haas Margolius (margolius(AT)math.csuohio.edu) [ [4]]
" " :1
111 :2
222 :10
333 :56
444 :346
555 :2252
etc... A000172 Franel number a(n) = Sum C(n,k)^3, k=0..n. [ [5]]
111 :3
222 :24
333 :216
444 :1824
555 :15150
etc... A000279 Card matching. [ [6]] COMMENT
Number of permutations of 3 distinct letters (ABC) each with n copies such that one (1) fixed points. E.g. if AAAAABBBBBCCCCC n=3*5 letters permutations then one fixed points n5=15150 - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Feb 02 2006
111 :0
222 :27
333 :378
444 :4536
555 :48600
etc... A000535 Card matching. [ [7]]
111 :1
222 :16
333 :435
444 :7136
555 :99350
etc... A000489 Card matching. [ [8]]
Zlajos 19. jun. 2007.
Zlajos 28. jun. 2007. 16. apr. 2009.
To show that for n ≥ 1, the expected number of fixed points is 1 :
We'll number the permutations p = 1 to n!
Now let X[p,m]=1 if in the p_th permutation, element m is fixed,
when it is not fixed, X[p,m]=0
Now the expected number of fixed points
is E_n[F] = sum_p_from_1_to_n! { sum_m_from_1_to_n { X[p,m] } } / n!
=> E_n[F] = sum_m_from_1_to_n { sum_p_from_1_to_n! { X[p,m] } } / n!
=> E_n[F] = sum_m_from_1_to_n { (n-1)! } / n!
=> E_n[F] = n * (n-1)! / n!
=> E_n[F] = 1
(with thanks to FD) Pnelnik ( talk) 08:42, 19 July 2009 (UTC)
On the article page the first table is not very pretty. There is no horizontal bar under number 2,3,4,5,6,7
and the verticle bar goes down too far:
0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | |
---|---|---|---|---|---|---|---|---|
0 | 1 | |||||||
1 | 0 | 1 |
Perhaps one solution would be to put in blanks in those extra cells.
0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | |
---|---|---|---|---|---|---|---|---|
0 | 1 | |||||||
1 | 0 | 1 |
It is not ideal, but I think it looks a bit better.
Pnelnik (
talk) 23:23, 19 July 2009 (UTC)
- This is the coefficient operator. Definition can be found at http://en.wikipedia.org/wiki/Formal_power_series#Extracting_coefficients Heycarnut ( talk) 08:37, 10 August 2013 (UTC)