login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo

Thanks to everyone who made a donation during our annual appeal!
To see the list of donors, or make a donation, see the OEIS Foundation home page.

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A329655 Square array read by antidiagonals: T(n,k) is the number of relations between set A with n elements and set B with k elements that are both right unique and left unique. 1
1, 2, 2, 3, 6, 3, 4, 12, 12, 4, 5, 20, 33, 20, 5, 6, 30, 72, 72, 30, 6, 7, 42, 135, 208, 135, 42, 7, 8, 56, 228, 500, 500, 228, 56, 8, 9, 72, 357, 1044, 1545, 1044, 357, 72, 9, 10, 90, 528, 1960, 4050, 4050, 1960, 528, 90, 10, 11, 110, 747, 3392, 9275, 13326, 9275, 3392, 747, 110, 11 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

A relation R between set A with n elements and set B with k elements is a subset of the Cartesian product A x B.  A relation R is right unique if (a, b1) in R and (a,b2) in R implies b1=b2.  A relation R is left unique if (a1,b) in R and (a2,b) in R implies a1=a2.

LINKS

Table of n, a(n) for n=1..66.

Roy S. Freedman, Some New Results on Binary Relations, arXiv:1501.01914 [cs.DM], 2015.

FORMULA

T(n,k) = Sum_{j=1..k} binomial(n,j)*binomial(k,j)*j!.

T(n,k) = A088699(n,k)-1.

EXAMPLE

The symmetric array T(n,k) begins:

  1,   2,    3,    4,     5,      6,       7,       8,        9, ...

  2,   6,   12,   20,    30,     42,      56,      72,       90, ...

  3,  12,   33,   72,   135,    228,     357,     528,      747, ...

  4,  20,   72,  208,   500,   1044,    1960,    3392,     5508, ...

  5,  30,  135,  500,  1545,   4050,    9275,   19080,    36045, ...

  6,  42,  228, 1044,  4050,  13326,   37632,   93288,   207774, ...

  7,  56,  357, 1960,  9275,  37632,  130921,  394352,  1047375, ...

  8,  72,  528, 3392, 19080,  93288,  394352, 1441728,  4596552, ...

  9,  90,  747, 5508, 36045, 207774, 1047375, 4596552, 17572113, ...

MAPLE

T:= (n, k)-> value(Sum(binomial(n, j)*binomial(k, j)*j!, j=1..k)):

seq(seq(T(n, 1+d-n), n=1..d), d=1..12);

MATHEMATICA

T[n_, k_] := Sum[Binomial[n, j] * Binomial[k, j] * j!, {j, 1, k}]; Table[T[n - k + 1, k], {n, 1, 11}, {k, 1, n}] // Flatten (* Amiram Eldar, Nov 25 2019 *)

PROG

(MuPAD) T:=(n, k)->_plus (binomial(n, j)*binomial(k, j)* j! $ j=1..k):

CROSSREFS

The diagonal T(n,n) is A097662.  T(1,k)=A000027; T(2,k)=A002378; T(3,k)=A054602.

Sequence in context: A125102 A003506 A047662 * A183474 A294034 A210220

Adjacent sequences:  A329652 A329653 A329654 * A329656 A329657 A329658

KEYWORD

nonn,tabl,easy

AUTHOR

Roy S. Freedman, Nov 18 2019

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified January 21 05:39 EST 2020. Contains 331104 sequences. (Running on oeis4.)