%I #7 Jul 11 2015 16:49:37
%S 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,
%T 9,45
%N Number parallelogram based on Pascal's triangle (and special mirror of central and multiply of diagonal).
%C Sequence arrangement:
%C ....................C(10,2)*C(0,0)
%C .............C(9,1)*C(1,1)...C(9,2)*C(1,0)
%C .....C(8,0)*C(2,2)...C(8,1)*C(2,1)...C(8,2)*C(2,0)
%C ..............C(7,0)*C(3,2)...C(7,1)*C(3,1)...C(7,2)*C(3,0)
%C .....................C(6,0)*C(4,2)...C(6,1)*C(4,1)...C(6,2)*C(4,0)
%C .............................C(5,0)*C(5,2)...C(5,1)*C(5,1)...C(5,2)*C(5,0)
%C .....................................C(4,0)*C(6,2)...C(4,1)*C(6,1)...C(4,2)*C(6,0)
%C .............................................C(3,0)*C(7,2)...C(3,1)*C(7,1)...C(3,2)*C(7,0)
%C .....................................................C(2,0)*C(8,2)...C(2,1)*C(8,1)...C(2,2)*C(8,0)
%C .............................................................C(1,0)*C(9,2)...C(1,1)*C(9,1)
%C .....................................................................C(0,0)*C(10,2)
%C "m" matching: analog (permutations with exactly "m" fixed points.
%C 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.)
%C if compared bbbbbbbbbb (b 10 times letters then 45 * "8" matching.(sum 45)
%C ("8" matching: analog (permutations with exactly 8 fixed points.).
%C If aabbbbbbbb (a 2 letters b 8 letters) permutations compared
%C aabbbbbbbb (a twice letters b 8 times letters)then 1 * "10"
%C matching),16 * "8" matching, 28 * "6" matching (sum 45)
%C If aabbbbbbbb (a 8 letters b 2 letters)permutations compared
%C aaaaaaaabb (a 8 times letters b twice letters)then 1 * "0"
%C matching),16 * "2" matching, 28 * "4" matching (sum 45)
%C all rows (sum 45)
%C etc...
%C matching equialent or analog "fixed points"
%C example:
%C arrangement relevant!
%C compared
%C letters..
%C times....
%C a...b
%C matching:..........0..1..2..3..4..5..6..7..8..9..10 0..10...................45.........................
%C .1..9.................9....36......................
%C .2..8..............1.....16....28..................
%C .3..7.................3....21.....21................
%C .4..6....................6....24....15............
%C .5..5......................10....25.....10.........
%C .6..4.........................15....24.....6........
%C .7..3.............................21....21.....3....
%C .8..2................................28....16......1
%C .9..1...................................36.....9....
%C 10..0......................................45.......
%C matching:...........0..1..2..3..4..5..6..7..8..9..10
%C The Maple code produces this:
%C 45, 36, 28, 21, 15, 10, 6, 3, 1
%C 9, 16, 21, 24, 25, 24, 21, 16, 9
%C 1, 3, 6, 10, 15, 21, 28, 36, 45
%C This is the table rotated right by Pi/4.
%H <a href="http://www.johnph77.com/math/lf.html#c4094">Lottery Numeric Positional Frequency Charts</a> 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) [From _Zerinvary Lajos_, Apr 02 2009]
%p 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;
%Y Cf. A113899.
%K fini,nonn
%O 0,1
%A _Zerinvary Lajos_, May 29 2007
%E Edited by _Charles R Greathouse IV_, Oct 28 2009