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 A113899 Number parallelogram based on Pascal's triangle (and special mirror of central and multiply of diagonal). 2

%I

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

%T 6,60,120,60,6,21,105,105,21,56,140,56,126,126,252

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

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

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

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

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

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

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

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

%C 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.)

%C 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)

%C 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.

%C etc...

%C matching equivalent "fixed-point"

%C example:

%C arrangement relevant!

%C compared

%C letters

%C times

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

%C compared.

%C letters..

%C times....

%C .a..b

%C 10..0.................................252..............................

%C .9..1...........................126.........126........................

%C .8..2......................56.........140..........56..................

%C .7..3................21.........105.........105..........21............

%C .6..4..........6...........60.........120..........60..........6.......

%C .5..5....1...........25.........100.........100..........25...........1

%C .4..6..........6...........60.........120..........60..........6.......

%C .3..7................21.........105.........105..........21............

%C .2..8......................56.........140..........56..................

%C .1..9...........................126.........126........................

%C 0..10..................................252.............................

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

%C The Maple code produces

%C 252, 126, 56, 21, 6, 1

%C 126, 140, 105, 60, 25, 6

%C 56, 105, 120, 100, 60, 21

%C 21, 60, 100, 120, 105, 56

%C 6, 25, 60, 105, 140, 126

%C 1, 6, 21, 56, 126, 252

%C which is the table rotated right by Pi/4.

%p 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

%Y Cf. A113162, A113163, A113164.

%K easy,fini,nonn,uned

%O 0,1

%A _Zerinvary Lajos_, Jan 29 2006, May 28 2007

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Last modified June 2 18:17 EDT 2020. Contains 334787 sequences. (Running on oeis4.)