A245792 -id:A245792 - 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

A245859 Number of preferential arrangements of n labeled elements such that the minimal number of elements per rank equals 6.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 924, 3432, 6006, 10010, 16016, 24752, 17190264, 139729800, 748339320, 2910015528, 9794896188, 30251595066, 2396910064472, 33228482071400, 291616291666700, 2036218597884900, 11895959650285620, 61536913327513260, 1662981928016982300

Offset: 6

Alois P. Heinz, Aug 04 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 6..400

Column k=6 of A245733.

a(n) = A245791, A245792, 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, 6) -b(n, 7):
seq(a(n), n=6..35);

E.g.f.: 1/(1-Sum_{j>=6} x^j/j!) - 1/(1-Sum_{j>=7} x^j/j!).

a(n) = A245791(n) - A245792(n) = A245732(n,6) - A245732(n,7).

A245860 Number of preferential arrangements of n labeled elements such that the minimal number of elements per rank equals 7.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 3432, 12870, 22880, 38896, 63648, 100776, 155040, 399305520, 3292693008, 17879790324, 70676513424, 242216077400, 762341522800, 2264840592300, 478970960616720, 6869326015894680, 61426122596911800, 435982960069722000, 2589856033041531072

Offset: 7

Alois P. Heinz, Aug 04 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 7..400

Column k=7 of A245733.

Cf. A245792, A245793, 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, 7) -b(n, 8):
seq(a(n), n=7..35);

E.g.f.: 1/(1-Sum_{j>=7} x^j/j!) - 1/(1-Sum_{j>=8} x^j/j!).

a(n) = A245792(n) - A245793(n) = A245732(n,7) - A245732(n,8).

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.

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A245859 Number of preferential arrangements of n labeled elements such that the minimal number of elements per rank equals 6.

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A245860 Number of preferential arrangements of n labeled elements such that the minimal number of elements per rank equals 7.

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