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%I #10 Jul 11 2015 10:34:13
%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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