A113899 Number parallelogram based on Pascal's triangle (and special mirror of central and multiply of diagonal).

252, 126, 126, 56, 140, 56, 21, 105, 105, 21, 6, 60, 120, 60, 6, 1, 25, 100, 100, 25, 1, 6, 60, 120, 60, 6, 21, 105, 105, 21, 56, 140, 56, 126, 126, 252

Offset: 0

Author

Zerinvary Lajos, Jan 29 2006, May 28 2007

Keywords

• easy
• fini
• nonn
• uned

Comments

.............................C(0,0)*C(10,5)

......................C(1,0)*C(9,5)...C(1,1)*C(9,4)

...............C(2,0)*C(8,5)...C(2,1)*C(8,4)...C(2,2)*C(8,3)

........C(3,0)*C(7,5)...C(3,1)*C(7,4)...C(3,2)*C(7,3)...C(3,3)*C(7,2)

...C(4,0)*C(6,5)...C(4,1)*C(6,4)...C(4,2)*C(6,3)...C(4,3)*C(8,2)...C(4,4)*C(6,1)

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

...C(6,1)*C(4,4)...C(4,1)*C(6,4)...C(4,2)*C(6,3)...C(4,3)*C(8,2)...C(6,5)*C(4,0)

........C(7,2)*C(3,3)...C(7,3)*C(3,2)...C(7,4)*C(3,1)...C(7,5)*C(3,0)

...............C(8,3)*C(2,2)...C(8,4)*C(2,1)...C(8,5)*C(2,0)

......................C(9,4)*C(1,1)...C(9,5)*C(1,0)

.............................C(10,5)*C(0,0)

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

if aaaaabbbbb (a 5 letters b 5 letters) permutations compared aaaaaaaaaa (a 10 times letters) or compared bbbbbbbbbb (b 10 times letters then 252 "5" matching. ("5" matching: analog (permutations with exactly 5 fixed points.)

If aaaaabbbbb (a 5 letters b 5 letters) permutations compared aaaaabbbbb (a 5 times letters b 5 times letters)then 1 "0" matching), 25 "2"matching 100 "4" matching, 100 "6" matching, 25 "8" matching and 1 "10" matching.(A008459 formatted as a triangular array: 6.rows)

If aaaaabbbbb (a 5 letters b 5 letters) permutations compared abbbbbbbbb (a 1 times letters b 9 times letters) or aaaaaaaaab (a 9 times letters b 1 times letters) then 126 "4" and 126 "6" matching.

etc...

matching equivalent "fixed-point"

example:

arrangement relevant!

compared

letters

times

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

compared.

letters..

times....

.a..b

10..0.................................252..............................

.9..1...........................126.........126........................

.8..2......................56.........140..........56..................

.7..3................21.........105.........105..........21............

.6..4..........6...........60.........120..........60..........6.......

.5..5....1...........25.........100.........100..........25...........1

.4..6..........6...........60.........120..........60..........6.......

.3..7................21.........105.........105..........21............

.2..8......................56.........140..........56..................

.1..9...........................126.........126........................

0..10..................................252.............................

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

The Maple code produces

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

which is the table rotated right by Pi/4.

Crossrefs

Cf. A113162, A113163, A113164.

Programs

Maple

for n from 0 to 5 do seq(binomial(i,n)*binomial(10-i,5-n), i=0+n..10-5+n ); # Zerinvary Lajos, Mar 31 2009