A245794 -id:A245794 - cp's OEIS frontend

A245732 Number T(n,k) of endofunctions on [n] such that at least one preimage with cardinality >=k exists and a nonempty preimage of j implies that all i<=j have preimages with cardinality >=k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

1, 1, 1, 4, 3, 1, 27, 13, 1, 1, 256, 75, 7, 1, 1, 3125, 541, 21, 1, 1, 1, 46656, 4683, 141, 21, 1, 1, 1, 823543, 47293, 743, 71, 1, 1, 1, 1, 16777216, 545835, 5699, 183, 71, 1, 1, 1, 1, 387420489, 7087261, 42241, 2101, 253, 1, 1, 1, 1, 1

Offset: 0

Alois P. Heinz, Jul 30 2014

T(0,0) = 1 by convention.

In general, column k > 1 is asymptotic to n! / ((1+r^(k-1)/(k-1)!) * r^(n+1)), where r is the root of the equation 2 - exp(r) + Sum_{j=1..k-1} r^j/j! = 0. - Vaclav Kotesovec, Aug 02 2014

			Triangle T(n,k) begins:
0 :         1;
1 :         1,      1;
2 :         4,      3,    1;
3 :        27,     13,    1,   1;
4 :       256,     75,    7,   1,  1;
5 :      3125,    541,   21,   1,  1, 1;
6 :     46656,   4683,  141,  21,  1, 1, 1;
7 :    823543,  47293,  743,  71,  1, 1, 1, 1;
8 :  16777216, 545835, 5699, 183, 71, 1, 1, 1, 1;

Alois P. Heinz, Rows n = 0..140, flattened

Column k=0 gives A000312.
Columns k=1-10 give (for n>0): A000670, A032032, A102233, A232475, A245790, A245791, A245792, A245793, A245794, A245795.
T(2n,n) gives A244174(n) or 1+A007318(2n,n) = 1+A000984(n) for n>0.
Cf. A245733.

b:= proc(n, k) option remember; `if`(n=0, 1,
      add(b(n-j, k)*binomial(n, j), j=k..n))
    end:
T:= (n, k)-> `if`(k=0, n^n, `if`(n=0, 0, b(n, k))):
seq(seq(T(n, k), k=0..n), n=0..12);

b[n_, k_] := b[n, k] = If[n == 0, 1, Sum[b[n-j, k]*Binomial[n, j], {j, k, n}]]; T[n_, k_] := If[k == 0, n^n, If[n == 0, 0, b[n, k]]]; T[0, 0] = 1; Table[Table[T[n, k], {k, 0, n}], {n, 0, 12}] // Flatten (* Jean-François Alcover, Jan 05 2015, after Alois P. Heinz *)

E.g.f. (for column k > 0): 1/(2 -exp(x) +Sum_{j=1..k-1} x^j/j!) -1. - Vaclav Kotesovec, Aug 02 2014

A245861 Number of preferential arrangements of n labeled elements such that the minimal number of elements per rank equals 8.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 12870, 48620, 87516, 151164, 251940, 406980, 639540, 980628, 9466982712, 78881427900, 432962644400, 1733914096200, 6029537213700, 19273224716460, 58178097911700, 168431757261300, 100033451495909100, 1461521434059544572

Offset: 8

Alois P. Heinz, Aug 04 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 8..400

Column k=8 of A245733.

Cf. A245793, A245794, A245732.

b:= proc(n, k) option remember; `if`(n=0, 1,
      add(b(n-j, k)*binomial(n, j), j=k..n))
    end:
a:= n-> b(n, 8) -b(n, 9):
seq(a(n), n=8..35);

E.g.f.: 1/(1-Sum_{j>=8} x^j/j!) - 1/(1-Sum_{j>=9} x^j/j!).

a(n) = A245793(n) - A245794(n) = A245732(n,8) - A245732(n,9).

A245862 Number of preferential arrangements of n labeled elements such that the minimal number of elements per rank equals 9.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 48620, 184756, 335920, 587860, 994840, 1634380, 2615008, 4085950, 6249100, 227882805150, 1914150638400, 10597377540750, 42894094729200, 150967391072550, 488846715676800, 1495608303532200, 4389524294884872, 12479799500904120

Offset: 9

Alois P. Heinz, Aug 04 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 9..400

Column k=9 of A245733.

Cf. A245794, A245795, A245732.

b:= proc(n, k) option remember; `if`(n=0, 1,
      add(b(n-j, k)*binomial(n, j), j=k..n))
    end:
a:= n-> b(n, 9) -b(n, 10):
seq(a(n), n=9..40);

E.g.f.: 1/(1-Sum_{j>=9} x^j/j!) - 1/(1-Sum_{j>=10} x^j/j!).

a(n) = A245794(n) - A245795(n) = A245732(n,9) - A245732(n,10).

cp's OEIS Frontend

A245732 Number T(n,k) of endofunctions on [n] such that at least one preimage with cardinality >=k exists and a nonempty preimage of j implies that all i<=j have preimages with cardinality >=k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A245861 Number of preferential arrangements of n labeled elements such that the minimal number of elements per rank equals 8.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Formula

A245862 Number of preferential arrangements of n labeled elements such that the minimal number of elements per rank equals 9.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Formula