A129536 Number parallelogram based on Pascal's triangle (and special mirror of central and multiply of diagonal).

45, 9, 36, 1, 16, 28, 3, 21, 21, 6, 24, 15, 10, 25, 10, 15, 24, 6, 21, 21, 3, 28, 16, 1, 36, 9, 45

Offset: 0

Author

Zerinvary Lajos, May 29 2007

Keywords

• fini
• nonn
• uned

Comments

Sequence arrangement:

....................C(10,2)*C(0,0)

.............C(9,1)*C(1,1)...C(9,2)*C(1,0)

.....C(8,0)*C(2,2)...C(8,1)*C(2,1)...C(8,2)*C(2,0)

..............C(7,0)*C(3,2)...C(7,1)*C(3,1)...C(7,2)*C(3,0)

.....................C(6,0)*C(4,2)...C(6,1)*C(4,1)...C(6,2)*C(4,0)

.............................C(5,0)*C(5,2)...C(5,1)*C(5,1)...C(5,2)*C(5,0)

.....................................C(4,0)*C(6,2)...C(4,1)*C(6,1)...C(4,2)*C(6,0)

.............................................C(3,0)*C(7,2)...C(3,1)*C(7,1)...C(3,2)*C(7,0)

.....................................................C(2,0)*C(8,2)...C(2,1)*C(8,1)...C(2,2)*C(8,0)

.............................................................C(1,0)*C(9,2)...C(1,1)*C(9,1)

.....................................................................C(0,0)*C(10,2)

"m" matching: analog (permutations with exactly "m" fixed points.

if aabbbbbbbb (a twice letters b 8 times letters) permutations compared aaaaaaaaaa (a 10 times letters) then 45 * "2" matching.(sum 45)("2" matching: analog(permutations with exactly 2 fixed points.)

if compared bbbbbbbbbb (b 10 times letters then 45 * "8" matching.(sum 45)

("8" matching: analog (permutations with exactly 8 fixed points.).

If aabbbbbbbb (a 2 letters b 8 letters) permutations compared

aabbbbbbbb (a twice letters b 8 times letters)then 1 * "10"

matching),16 * "8" matching, 28 * "6" matching (sum 45)

If aabbbbbbbb (a 8 letters b 2 letters)permutations compared

aaaaaaaabb (a 8 times letters b twice letters)then 1 * "0"

matching),16 * "2" matching, 28 * "4" matching (sum 45)

all rows (sum 45)

etc...

matching equialent or analog "fixed points"

example:

arrangement relevant!

compared

letters..

times....

a...b

matching:..........0..1..2..3..4..5..6..7..8..9..10 0..10...................45.........................

.1..9.................9....36......................

.2..8..............1.....16....28..................

.3..7.................3....21.....21................

.4..6....................6....24....15............

.5..5......................10....25.....10.........

.6..4.........................15....24.....6........

.7..3.............................21....21.....3....

.8..2................................28....16......1

.9..1...................................36.....9....

10..0......................................45.......

matching:...........0..1..2..3..4..5..6..7..8..9..10

The Maple code produces this:

45, 36, 28, 21, 15, 10, 6, 3, 1

9, 16, 21, 24, 25, 24, 21, 16, 9

1, 3, 6, 10, 15, 21, 28, 36, 45

This is the table rotated right by Pi/4.

Links

• Johnph77, Lottery Numeric Positional Frequency Charts Note:Information herein is intended for lottery system developers, analysts and operators. It is not intended for gaming purposes. 3/11 table: (Horizontal > Total: 165, Vertical > Total: 45)

Crossrefs

Cf. A113899.

Programs

Maple

with(combinat):T:=(n,i)->binomial(i,n)*binomial(10-i,2-n): for n from 0 to 2 do seq(T(n, i), i=0+n..10-2+n) od;

Extensions

Edited by Charles R Greathouse IV, Oct 28 2009