# Talk:Derangement

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d(n) also satisfies the recurrence: d(n) = n*d(n-1) + (-1)^n.
see: http://mathworld.wolfram.com/Derangement.html
I've seen the relation proved by inclusion-exclusion.
DonkeyKong the mathematician (in training) 08:19, 17 July 2006 (UTC)

## General formula

The page about rencontres numbers gives a general formula for the derangements as the closest integer to ${\displaystyle n! \over e}$. Is this formula valid for all n? I checked with the first points (ok), and it's obviously valid in the limit. Is there a proof? A corollary of this formula would be a proof that e is irrational. Albmont 13:24, 14 November 2006 (UTC)

It is the closest for all natural numbers except n=0. You can always get the correct integer for dn by rounding up at even n and rounding down at odd n. JRSpriggs 05:10, 23 December 2006 (UTC)
Probably the formula ${\displaystyle !n=\left\lfloor {\frac {n!}{e}}+{\frac {1}{2}}\right\rfloor }$ should somehow state this, that it doesn't work for n=0?--Thomasda (talk) 15:06, 28 October 2010 (UTC)
Yeah, of course, I added it to the article. Paul Breeuwsma (talk) 15:25, 5 December 2010 (UTC)

START Zlajos 17 jun 2007

## 1.PART

Extension: If all character once : example: ABCDE......

• A008290 Triangle T(n,k) of rencontres numbers (number of *permutations of n elements with k fixed points).[[1]]
The proof is an application of the inclusion-exclusion principle. I'm a bit surprised that this page doesn't say that. Michael Hardy 16:44, 19 June 2007 (UTC)
The details are on the page for Random Permutation Statistics], Maybe we should link there? -Zahlentheorie 09:45, 20 June 2007 (UTC)

## 1.table

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
• If all character twice: example: AABBCC....
• A059056 Penrice Christmas gift numbers, Card-matching numbers (Dinner-Diner matching numbers). [[2]]

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

## 2.table

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)

• "0" (table sign: "0")then 1 derangements,
• "A" (table sign: 1)then 0 derangements,
• "AB" (table sign: 11)then 1 derangements,
• "ABC" (table sign: 111)then 2 derangements,
• "ABCD" (table sign: 1111)then 9 derangements, etc.

column > free or 0 :

• 1, 0, 1, 2, 9, 44, 265, 1854, 14833, 133496, 1334961...
• 00166 Subfactorial or rencontres numbers, or derangements: number of permutations of n elements with no fixed points.[[00166 Subfactorial or rencontres numbers, or derangements: number of permutations of n elements with no fixed points.]]

then:

• analogous (2.table)
• "0" (table sign: "0")then 1 derangements,
• AA (table sign: 2)then 0 derangements,
• AABB (table sign: 22)then 1 derangements,
• AABBCC (table sign: 222)then 10 derangements,
• AABBCCDD (table sign: 2222)then 297 derangements, etc.
• column > free or 0 :
• 1, 0, 1, 10, 297, 13756, 925705, 85394646,...
• A059072 Penrice Christmas gift numbers; card-matching numbers; dinner-diner matching numbers.[[3]]

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) )

• COMMENT Number of fixed-point-free permutations of n distinct letters (ABCD...), each of which appears twice. If there is only one letter of each type we get A000166. - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Oct 15 2006

• Question:
• 2.table
• column: 2,3,4,5,...
• where is it :formula or generating function(?)
• where is it :bibliography?

## 3.table

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)

• "0" (table sign: "0")then 1 derangements,
• "A" (table sign: 1)then 0 derangements,
• "AB" (table sign: 11)then 1 derangements,
• "ABC" (table sign: 111)then 2 derangements,
• "ABCD" (table sign: 1111)then 9 derangements, etc.
• column > free or 0 :
• 1, 0, 1, 2, 9, 44, 265, 1854, 14833, 133496, 1334961...
• 00166 Subfactorial or rencontres numbers, or derangements: number of permutations of n elements with no fixed points.[[00166 Subfactorial or rencontres numbers, or derangements: number of permutations of n elements with no fixed points.]]

then:

• analogous (3.table)
• "0" (table sign: "0")then 1 derangements,
• AAA (table sign: 3)then 0 derangements,
• AAABBB (table sign: 33)then 1 derangements,
• AAABBBCCC (table sign: 333)then 56 derangements,
• AAABBBCCCDDD (table sign: 3333)then 13833 derangements, etc.
• column > free or 0 :
• 1, 0, 1, 56, 13833, 6699824, 5691917785, 7785547001784,
• A059073 Card-matching numbers (Dinner-Diner matching numbers).

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]]

• Number of fixed-point-free permutations of n distinct letters (ABCD...), each of which appears thrice. If there is only one letter of each type we get A000166. - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Oct 15 2006

• 2.column (free or "0" -fixed point

" "  :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]]

• 3.column ( "1" -fixed point)

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

• 4.column ( "2" fixed point)

111  :0

222  :27

333  :378

444  :4536

555  :48600

etc... A000535 Card matching. [[7]]

• 5.column ( "3" fixed point)

111  :1

222  :16

333  :435

444  :7136

555  :99350

etc... A000489 Card matching. [[8]]

• 3.table
• column: 2,3,4,5,...
• where is it :formula or generating function(?)
• where is it :bibliography?

continued:

• charcters:quadruple, example:AAAA, AAAABBBB, AAAABBBBCCCC, AAAABBBBCCCCDDDD, etc...
• table 1.column :4, 44, 444, 4444, 44444, etc...
• charcters:quintuple, example:AAAAA, AAAAABBBBB, AAAAABBBBBCCCCC, etc...
• table 1.column :5, 55, 555, 5555, 55555, etc...
• a great number of connexion of interesting !!
Zlajos


19. jun. 2007.

Zlajos 21. jun. 2007. Extension: If all character twice : example: AABBCC, which has 2 A, 2 B's, and 2 C's, is

${\displaystyle {6 \choose 2,2,2}={\frac {6!}{2!\,2!\,2!}}=90}$

Compare the all distinct anagram for AABBCC to CCBBAA (90) one after the other :template (or schema)

AAAAAA or 6 0 0 equal, (identical): BBBBBB and CCCCCC

AAAAAB or 5 1 0 equal, (identical): BBBBBC and CCCCCA etc.

AAAABB or 4 2 0 equal, (identical): AAAACC and BBBBAA etc.

AAAABC or 4 1 1 equal, (identical): CCCCAB and BBBBAC etc.

AAABBB or 3 3 0 equal, (identical): AAACCC and BBBCCC etc.

AABBCC or 2 2 2

AAABBC or 3 2 1 equal, (identical): BBBCCA and CCCAAB etc.

## 4.table

fixed point: character numbers: free or "0" 1 2 3 4 5 6 sum
6 0 0 or AAAAAA 0 0 90 0 0 0 0 90
5 1 0 or AAAAAB 0 30 30 30 0 0 0 90
4 2 0 or AAAABB 6 24 30 24 6 0 0 90
4 1 1 or AAAABC 6 24 36 12 12 0 0 90
3 3 0 or AAABBB 9 18 36 18 9 0 0 90
2 2 2 or AABBCC 10 24 27 16 12 0 1 90
3 2 1 or AAABBC 12 27 33 15 3 0 0 90

Extension: If all character thrice : example: AAABBBCCC, which has 3 A, 3 B's, and 3 C's, is

${\displaystyle {9 \choose 3,3,3}={\frac {9!}{3!\,3!\,3!}}=1680}$

Compare the all distinct anagram for AAABBBCCC to CCCBBBAAA (1680) one after the other :template (or schema)

AAAAAAAAA or 9 0 0 equal, (identical): BBBBBBBBB and CCCCCCCCC

AAAAAAAAB or 8 1 0 equal, (identical): BBBBBBBBC and CCCCCCCCA etc.

AAAAAAABB or 7 2 0 equal, (identical): AAAAAAACC and BBBBBBBAA etc.

AAAAAAABC or 7 1 1 equal, (identical): CCCCCCCAB and BBBBBBBAC etc.

AAAAAABBB or 6 3 0 equal, (identical): AAAAAACCC and BBBBBBCCC etc.

AAAAAABBC or 6 2 1 equal, (identical): AAAAAACCB and BBBBBBCCA etc. .................... AAABBBCCC or 3 3 3 etc...

## 5.table

fixed point: character numbers: free or "0" 1 2 3 4 5 6 7 8 9 sum
9 0 0 or AAAAAAAAA 0 0 0 1680 0 0 0 0 0 0 1680
8 1 0 or AAAAAAAAB 0 0 560 560 560 0 0 0 0 0 1680
7 2 0 or AAAAAAABB 0 140 420 560 420 140 0 0 0 0 1680
7 1 1 or AAAAAAABC 0 140 420 630 280 210 0 0 0 0 1680
6 3 0 or AAAAAABBB 20 180 360 560 360 180 20 0 0 0 1680
6 2 1 or AAAAAABBC 20 180 420 480 380 140 60 0 0 0 1680
5 4 0 or AAAAABBBB 40 160 400 480 400 160 40 0 0 0 1680
5 3 1 or AAAAABBBC 40 190 400 460 360 160 60 10 0 0 1680
5 2 2 or AAAAABBCC 40 200 400 460 320 200 40 20 0 0 1680
4 4 1 or AAAABBBBC 48 192 384 480 320 192 48 16 0 0 1680
4 3 2 or AAAABBBCC 52 208 388 436 340 168 72 12 4 0 1680
3 3 3 or AAABBBCCC 56 216 378 435 324 189 54 27 0 1 1680

...4. table, 5.table sum: 90, 1680, etc.:A006480 De Bruijn's s(3,n): (3n)!/(n!)^3. [[10]]

continued! Zlajos 28. jun. 2007.

Compare the all distinct anagram for AAAAAABBBBBBB to BBBBBBAAAAAA (924) one after the other :template (or schema)

one after the other :template (or schema)

AAAAAAAAAAAA or 12 0

AAAAAAAAAAAB or 11 1

AAAAAAAAAABB or 10 2

....................

....................

BBBBBBBBBBAA or 2 10

....................

BBBBBBBBBBBB or 0 12

analogous or similar: A129352 [[11]]

MAPLE:with(combinat):T:=(n,i)->binomial(i,n)*binomial(12-i,6-n): for n from 0 to 6 do seq(T(n, i), i=0+n..12-6+n) od; #Warning, new definition for Chi

924, 462, 210, 84, 28, 7, 1

462, 504, 378, 224, 105, 36, 7

210, 378, 420, 350, 225, 105, 28

84, 224, 350, 400, 350, 224, 84

28, 105, 225, 350, 420, 378, 210

7, 36, 105, 224, 378, 504, 462

1, 7, 28, 84, 210, 462, 924

If this is table rotated right by Pi/4. then equal 6.table

## 6.table

fixed point: character numbers: "0" 1 2 3 4 5 6 7 8 9 10 11 12 sum
12 0 or AAAAAAAAAAAA 924 924
11 1 or AAAAAAAAAAAB 462 462 924
10 2 or AAAAAAAAAABB 210 504 210 924
9 3 or AAAAAAAAABBB 84 378 378 84 924
8 4 or AAAAAAAABBBB 28 224 420 224 28 924
7 5 or AAAAAAABBBBB 7 105 350 350 105 7 924
6 6 or AAAAAABBBBBB 1 36 225 400 225 36 1 924
5 7 or AAAAABBBBBBB 7 105 350 350 105 7 924
4 8 or AAAABBBBBBBB 28 224 420 224 28 924
3 9 or AAABBBBBBBBB 84 378 378 84 924
2 10 or AABBBBBBBBBB 210 504 210 924
1 11 or ABBBBBBBBBBB 462 462 924
0 12 or BBBBBBBBBBBB 924 924

## 7.table

fixed point: character numbers: "0" 1 2 3 4 5 6 7 8 9 10 11 12 sum
12 0 or AAAAAAAAAAAA C(0,0)*C(12,6) 924
11 1 or AAAAAAAAAAAB C(1,0)*C(11,6) C(1,1)*C(11,5) 924
10 2 or AAAAAAAAAABB C(2,0)*C(10,6) C(2,1)*C(10,5 C(2,2)*C(10,4) 924
9 3 or AAAAAAAAABBB C(3,0)*C(9,6) C(3,1)*C(9,5) C(3,2)*C(9,4) C(3,3)*C(9,3) 924
8 4 or AAAAAAAABBBB C(4,0)*C(8,6) C(4,1)*C(8,5) C(4,2)*C(8,4) C(4,3)*C(8,3) C(4,4)*C(8,2) 924
7 5 or AAAAAAABBBBB C(5,0)*C(7,6) C(5,1)*C(7,5) C(5,2)*C(7,4) C(5,3)*C(7,3) C(5,4)*C(7,2) C(5,5)*C(7,1) 924
6 6 or AAAAAABBBBBB C(6,0)*C(6,6) C(6,1)*C(6,5) C(6,2)*C(6,4) C(6,3)*C(6,3) C(6,4)*C(6,2) C(6,5)*C(6,1) C(6,6)*C(6,0) 924
5 7 or AAAAABBBBBBB C(7,1)*C(5,5) C(7,2)*C(5,4) C(7,3)*C(5,3) C(7,4)*C(5,2) C(7,5)*C(5,1) C(7,6)*C(5,0) 924
4 8 or AAAABBBBBBBB C(8,2)*C(4,4) C(8,3)*C(4,3) C(8,4)*C(4,2) C(8,5)*C(4,1) C(8,6)*C(4,0) 924
3 9 or AAABBBBBBBBB C(9,3)*C(3,3) C(9,4)*C(3,2) C(9,5)*C(3,1) C(9,6)*C(3,0) 924
2 10 or AABBBBBBBBBB C(10,4)*C(2,2) C(10,5)*C(2,1) C(10,6)*C(2,0) 924
1 11 or ABBBBBBBBBBB C(11,5)*C(1,1) C(11,6)*C(1,0) 924
0 12 or BBBBBBBBBBBB C(12,6)*C(0,0) 924

## 8.table

PASCAL TRIANGLE item, (portion)

fixed point: character numbers: "0" 1 2 3 4 5 6 7 8 9 10 11 12
............................ C(0,0)*
C(1,0)* C(1,1)*
C(2,0)* C(2,1)* C(2,2)*
C(3,0)* C(3,1)* C(3,2)* C(3,3)*
C(4,0)* C(4,1)* C(4,2)* C(4,3)* C(4,4)*
C(5,0)* C(5,1)* C(5,2)* C(5,3)* C(5,4)* C(5,5)*
C(6,0)* C(6,1)* C(6,2)* C(6,3)*

centre

C(6,4)* C(6,5)* C(6,6)* ..................

## 9.table

 ............................. *C(6,6) *C(6,5) *C(6,4) *C(6,3) centre *C(6,2) *C(6,1) *C(6,0) ................. . *C(5,5) *C(5,4) *C(5,3) *C(5,2) *C(5,1) *C(5,0) . *C(4,4) *C(4,3) *C(4,2) *C(4,1) *C(4,0) . *C(3,3) *C(3,2) *C(3,1) *'C(3,0)' . *C(2,2) *C(2,1) *C(2,0) . *C(1,1) *C(1,0) . *C(0,0)

continued! Zlajos 04. jul. 2007.

## 2. PART

Maple list:

for n from 0 to 0 do seq(binomial(i,n)*binomial(2-i,0-n), i=0+n..2-0+n ); od;#

for n from 0 to 1 do seq(binomial(i,n)*binomial(2-i,1-n), i=0+n..1-0+n ); od;#

for n from 0 to 2 do seq(binomial(i,n)*binomial(4-i,2-n), i=0+n..4-2+n ); od;#

for n from 0 to 3 do seq(binomial(i,n)*binomial(6-i,3-n), i=0+n..6-3+n ); od;

for n from 0 to 4 do seq(binomial(i,n)*binomial(8-i,4-n), i=0+n..8-4+n ); od;

for n from 0 to 5 do seq(binomial(i,n)*binomial(10-i,5-n), i=0+n..10-5+n );od

for n from 0 to 6 do seq(binomial(i,n)*binomial(12-i,6-n), i=0+n..12-6+n ); od;#

for n from 0 to 7 do seq(binomial(i,n)*binomial(14-i,7-n), i=0+n..14-7+n ); od;#

for n from 0 to 8 do seq(binomial(i,n)*binomial(16-i,8-n), i=0+n..16-8+n ); od;#

for n from 0 to 9 do seq(binomial(i,n)*binomial(18-i,9-n), i=0+n..18-9+n ); od;#

for n from 0 to 10 do seq(binomial(i,n)*binomial(20-i,10-n), i=0+n..20-10+n ); od;#

## To simplify table (simple table): for 1 to 8

• 0.

1, 1, 1

• 1.

2, 1

1, 2

• 2.

6, 3, 1

3, 4, 3

1, 3, 6

• 3.

20, 10, 4, 1

10, 12, 9, 4

4, 9, 12, 10

1, 4, 10, 20

• 4.

70, 35, 15, 5, 1

35, 40, 30, 16, 5

15, 30, 36, 30, 15

5, 16, 30, 40, 35

1, 5, 15, 35, 70

• 5.

252, 126, 56, 21, 6, 1

126, 140, 105, 60, 25, 6

56, 105, 120, 100, 60, 21

21, 60, 100, 120, 105, 56

6, 25, 60, 105, 140, 126

1, 6, 21, 56, 126, 252

• 6.

924, 462, 210, 84, 28, 7, 1

462, 504, 378, 224, 105, 36, 7

210, 378, 420, 350, 225, 105, 28

84, 224, 350, 400, 350, 224, 84

28, 105, 225, 350, 420, 378, 210

7, 36, 105, 224, 378, 504, 462

1, 7, 28, 84, 210, 462, 924

• 7.

3432, 1716, 792, 330, 120, 36, 8, 1

1716, 1848, 1386, 840, 420, 168, 49, 8

792, 1386, 1512, 1260, 840, 441, 168, 36

330, 840, 1260, 1400, 1225, 840, 420, 120

120, 420, 840, 1225, 1400, 1260, 840, 330

36, 168, 441, 840, 1260, 1512, 1386, 792

8, 49, 168, 420, 840, 1386, 1848, 1716

1, 8, 36, 120, 330, 792, 1716, 3432

• 8.

12870, 6435, 3003, 1287, 495, 165, 45, 9, 1

6435, 6864, 5148, 3168, 1650, 720, 252, 64, 9

3003, 5148, 5544, 4620, 3150, 1764, 784, 252, 45

1287, 3168, 4620, 5040, 4410, 3136, 1764, 720, 165

495, 1650, 3150, 4410, 4900, 4410, 3150, 1650, 495

165, 720, 1764, 3136, 4410, 5040, 4620, 3168, 1287

45, 252, 784, 1764, 3150, 4620, 5544, 5148, 3003

9, 64, 252, 720, 1650, 3168, 5148, 6864, 6435

1, 9, 45, 165, 495, 1287, 3003, 6435, 12870

etc...

## 3. PART

all 1.rows 1. numbers (and mirror)

1, 2, 6, 20, 70, 252, 924, 3432, 12870, etc...

Central binomial coefficients: C(2n,n) = (2n)!/(n!)^2.

A000984[[12]]

all 1.rows 2. numbers (and mirror)

1, 3, 10, 35, 126, 462, 1716, 6435, 24310, 92378, 352716, etc...

C(2n+1, n+1)

A001700 [[13]]

all 1.rows 3. numbers (and mirror)

1, 4, 15, 56, 210, 792, 3003, 11440, 43758, 167960, etc...

• Binomial coefficients C(2n,n-1).

A001791 [[14]]

all 1.rows 4. numbers (and mirror)

1, 5, 21, 84, 330, 1287, 5005, 19448, 75582, etc...

• Binomial coefficient binomial(2n+1,n-1).

A002054 [[15]]

all 1.rows 5. numbers (and mirror)

1, 7, 36, 165, 715, 3003, 12376, 50388, 203490, 817190, etc...

• Binomial coefficients C(2n+1,n-2).

A003516 [[16]]

all 1.rows 6. numbers (and mirror)

1, 7, 36, 165, 715, 3003, 12376, 50388, 203490, etc...

• Binomial coefficients C(2n+1,n-2).

A003516[[17]]

all 1.rows 7. numbers (and mirror)

1, 8, 45, 220, 1001, 4368, 18564, 77520, 319770, etc...

• Binomial coefficients C(2n,n-3).

A002696 [[18]]

all 2.rows 1. numbers (and mirror)equal all 1.rows 2. numbers

all 2.rows 2. numbers (and mirror)

2, 4, 12, 40, 140, 504, 1848, 6864, 25740, 97240, etc...

Twice central binomial coefficients

A028329[[19]]

all 2.rows 3. numbers (and mirror)

3, 9, 30, 105, 378, 1386, 5148, 19305, 72930, 277134, 1058148,etc...

3*C(2*n-1,n).

A003409 [[20]]

all 3.rows 3. numbers (and mirror)

6, 12, 36, 120, 420, 1512, 5544 etc...

A067804 formatted as a square array:3.rows [[21]]

all 4.rows 4. numbers (and mirror)

20, 40, 120, 400, 1400, 5040, etc...

A067804 formatted as a square array:4.rows [[22]]

all 5.rows 5. numbers (and mirror)

70, 140, 420, 1400, 4900,etc...

A067804 formatted as a square array:5.rows [[23]]

etc...

etc...

A067804 formatted as a square array:

1 2 6 20 70 252 924 3432 12870

2 4 12 40 140 504 1848 6864

6 12 36 120 420 1512 5544

20 40 120 400 1400 5040

70 140 420 1400 4900

252 504 1512 5040

924 1848 5544

3432 6864

12870

...................................................

all diagonal left to right and bottom to top

Square the entries of Pascal's triangle.

A008459 [[24]]

all 2.table "center" 1, 4, 36, 400, 4900, 63504, 853776, etc...

• Binomial(2n,n)^2.

A002894 [[25]]

Everything to correlate everything....

I am search: bibliography (internet), proof and etc...

continued! Zlajos 05. jul. 2007.

## Not speak English

Sorry, not speak english. All corrections thanks! Zerinvary Lajos, Hungary Zlajos OEIS>>zerinvarylajos or zerinvary >>e-mail — Preceding unsigned comment added by 78.92.185.226 (talk) 06:48, 16 April 2009 (UTC)

## Number parallelogram based on Pascal's triangle (and special mirror of central and multiply of diagonal)

OEIS
1. A113899 >>[26]
2. A129352 >> [27]
3. A129536 >> [28]
4. Demo>>...mirror of central and multiply of diagonal...[29] (Pascal háromszög tükrözése és szorzás. Minta.)

—Preceding unsigned comment added by Zlajos (talkcontribs) 07:03, 16 April 2009 (UTC)

## Applications

Would it be possible to add a section on why this is useful, what are the applications of this math? - Thanks. — Preceding unsigned comment added by 62.60.15.67 (talk) 08:53, 28 August 2012 (UTC)

## Dual Dearrangement

Suppose we have a list A of 2n distinct elements {a1i, a2i; i=1, 2, ..., n}. A dual dearrangement is a permutation of A in which no element a1i or a2i is at any positions 2i and 2i-1, and no two elements a1i and a2i occupy the positions 2j and 2j-1, for any j, in any order.

The method used to derive derangement problem doesn't seem to work for the dual derangement. Any thought?

Mileszhou (talk) 08:12, 20 March 2014 (UTC) Miles Zhou