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A181386
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Tetrahedron of terms C(r,n,m) representing the number of ways of choosing m disjoint subsets of r members from an original set of n members.
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0
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1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 3, 1, 1, 1, 1, 1, 1, 4, 6, 4, 1, 1, 3, 1, 1, 1, 1, 5, 10, 10, 5, 1, 1, 6, 3, 1, 1, 1, 1, 1, 1, 6, 15, 20, 15, 6, 1, 1, 10, 15, 1, 4, 1, 1, 1, 1, 1, 7, 21, 35, 35, 21, 7, 1, 1, 15, 45, 15, 1, 10, 1, 1, 1, 1, 1, 1, 1, 8, 28, 56, 70, 56, 28, 8, 1, 1, 21, 105, 105, 1, 20
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OFFSET
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1,6
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COMMENTS
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The start index for r is 1 but the start index for m and n is 0. For each value of r, the triangle T_r(n,m) has row n containing 1+floor(n/r) terms.
Contribution from Frank M Jackson, Nov 20 2010: (Start)
C(r,mr,m)=C(r,mr-1,m-1).
C(1,m,m)=A000012, C(2,2m,m)=A001147,
C(3,3m,m),...,C(10,10m,m)=A025035,...,A025042.
C(2,26,10)=150,738,274,937,250 and represents the number of possible plugboard settings for a WWII German Enigma Enciphering Machine.
C(r,2r,2)=A001700, C(r,3r,3)=A060542, C(r,4r,4)=A082368.
C(r,n,m)=C(r,mr-1,m-1)*Binomial[n,rm],
and applied recursively gives identity
C(r,n,m)=Binomial[n,r*m]*Product[Binomial[r(m-p+1)-1,r-1],{p,m}].
(End)
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LINKS
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Table of n, a(n) for n=1..92.
T. Copeland, Infinitesimal Generators, the Pascal Pyramid, and the Witt and Virasoro Algebras
Tony Sale,Possible Plugboard Settings for a WWII German Enigma Enciphering Machine
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FORMULA
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C(r,n,m)=n!/((n-r*m)!*m!*(r!)^m
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EXAMPLE
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r=1, C(1,n,m) is
1
1,1
1,2,1
1,3,3,1
1,4,6,4,1
1,5,10,10,5,1
r=2, C(2,n,m) is
1
1
1,1
1,3
1,6,3
1,10,15
r=3, C(3,n,m) is
1
1
1
1,1
1,4
1,10
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MATHEMATICA
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Flatten[Table[{n!/((n-r*m)!*m!*r!^m)}, {r, 1, 50}, {n, 0, 50}, {m, 0, Floor[n/r]}]]
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CROSSREFS
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C(1, n, m)=T_1(n, m)=A007318 and C(2, n, m)=T_2(n, m)=A100861
Sequence in context: A026536 A046213 A215625 * A193517 A189006 A219924
Adjacent sequences: A181383 A181384 A181385 * A181387 A181388 A181389
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KEYWORD
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nonn,tabf
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AUTHOR
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Frank M Jackson, Oct 16 2010
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STATUS
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approved
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