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.

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

Offset: 1

Author

Frank M Jackson, Oct 16 2010

Keywords

• nonn
• tabf

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.

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) = 150738274937250 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 the identity

C(r,n,m) = Binomial(n,r*m) * Product_{p=1..m} Binomial(r*(m-p+1)-1,r-1).

(End)

C(2,26,10) = A266365(10), where 26 is the size of the alphabet. - Jonathan Sondow, Dec 29 2015

Examples

			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
		

Links

• 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

Crossrefs

C(1,n,m) = T_1(n,m) = A007318, C(2,n,m) = T_2(n,m) = A100861, and C(2,26,m) = A266365.

Programs

Mathematica

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

Formula

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