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A181386 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. 0
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,6

COMMENTS

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)

LINKS

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

FORMULA

C(r,n,m)=n!/((n-r*m)!*m!*(r!)^m

EXAMPLE

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

MATHEMATICA

Flatten[Table[{n!/((n-r*m)!*m!*r!^m)}, {r, 1, 50}, {n, 0, 50}, {m, 0, Floor[n/r]}]]

CROSSREFS

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 A245013

Adjacent sequences:  A181383 A181384 A181385 * A181387 A181388 A181389

KEYWORD

nonn,tabf

AUTHOR

Frank M Jackson, Oct 16 2010

STATUS

approved

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Last modified December 18 23:18 EST 2014. Contains 252175 sequences.