A113163 -id:A113163 - cp's OEIS frontend

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

Zerinvary Lajos, Jan 29 2006, May 28 2007

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

Cf. A113162, A113163, A113164.

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

cp's OEIS Frontend

A113162 a(n) = binomial(6, n) * binomial(7, 6-n).

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