A262071
Number T(n,k) of ordered partitions of an n-set with nondecreasing block sizes and maximal block size equal to k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
Original entry on oeis.org
1, 0, 1, 0, 2, 1, 0, 6, 3, 1, 0, 24, 18, 4, 1, 0, 120, 90, 30, 5, 1, 0, 720, 630, 200, 45, 6, 1, 0, 5040, 4410, 1610, 350, 63, 7, 1, 0, 40320, 37800, 13440, 3290, 560, 84, 8, 1, 0, 362880, 340200, 130200, 30870, 5922, 840, 108, 9, 1, 0, 3628800, 3515400, 1327200, 334950, 61992, 9870, 1200, 135, 10, 1
Offset: 0
T(3,1) = 6: 1|2|3, 1|3|2, 2|1|3, 2|3|1, 3|1|2, 3|2|1.
T(3,2) = 3: 1|23, 2|13, 3|12.
T(3,3) = 1: 123.
Triangle T(n,k) begins:
1;
0, 1;
0, 2, 1;
0, 6, 3, 1;
0, 24, 18, 4, 1;
0, 120, 90, 30, 5, 1;
0, 720, 630, 200, 45, 6, 1;
0, 5040, 4410, 1610, 350, 63, 7, 1;
0, 40320, 37800, 13440, 3290, 560, 84, 8, 1;
Columns k=0-10 give:
A000007,
A000142 (for n>0),
A272492,
A272493,
A272494,
A272495,
A272496,
A272497,
A272498,
A272499,
A272500.
-
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
b(n, i-1)+`if`(i>n, 0, binomial(n, i)*b(n-i, i))))
end:
T:= (n, k)-> b(n, k) -`if`(k=0, 0, b(n, k-1)):
seq(seq(T(n, k), k=0..n), n=0..12);
-
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, b[n, i - 1] + If[i > n, 0, Binomial[n, i]*b[n - i, i]]]]; T[n_, k_] := b[n, k] - If[k == 0, 0, b[n, k - 1]]; Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-François Alcover, Dec 12 2016, Alois P. Heinz *)
A327801
Sum T(n,k) of multinomials M(n; lambda), where lambda ranges over all partitions of n into parts incorporating k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
Original entry on oeis.org
1, 1, 1, 3, 2, 1, 10, 9, 3, 1, 47, 40, 18, 4, 1, 246, 235, 100, 30, 5, 1, 1602, 1476, 705, 200, 45, 6, 1, 11481, 11214, 5166, 1645, 350, 63, 7, 1, 95503, 91848, 44856, 13776, 3290, 560, 84, 8, 1, 871030, 859527, 413316, 134568, 30996, 5922, 840, 108, 9, 1
Offset: 0
Triangle T(n,k) begins:
1;
1, 1;
3, 2, 1;
10, 9, 3, 1;
47, 40, 18, 4, 1;
246, 235, 100, 30, 5, 1;
1602, 1476, 705, 200, 45, 6, 1;
11481, 11214, 5166, 1645, 350, 63, 7, 1;
95503, 91848, 44856, 13776, 3290, 560, 84, 8, 1;
...
-
with(combinat):
T:= (n, k)-> add(multinomial(add(i, i=l), l[], 0), l=
select(x-> k=0 or k in x, partition(n))):
seq(seq(T(n, k), k=0..n), n=0..10);
# second Maple program:
b:= proc(n, i, k) option remember; `if`(n=0, 1,
`if`(i<2, 0, b(n, i-1, `if`(i=k, 0, k)))+
`if`(i=k, 0, b(n-i, min(n-i, i), k)/i!))
end:
T:= (n, k)-> n!*(b(n$2, 0)-`if`(k=0, 0, b(n$2, k))):
seq(seq(T(n, k), k=0..n), n=0..10);
-
b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 2, 0, b[n, i - 1, If[i == k, 0, k]]] + If[i == k, 0, b[n - i, Min[n - i, i], k]/i!]];
T[n_, k_] := n! (b[n, n, 0] - If[k == 0, 0, b[n, n, k]]);
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Apr 30 2020, from 2nd Maple program *)
A328156
Number of set partitions of [2n] with distinct block sizes and one of the block sizes is n.
Original entry on oeis.org
1, 0, 0, 60, 280, 3780, 74844, 576576, 6949800, 110416020, 3319141540, 31333878576, 545777101324, 8349081650000, 196469122903200, 8108831645948160, 99934219113287400, 1961077012271694900, 39215221761564594900, 860948656518718429200, 25274389422461123124180
Offset: 0
-
b:= proc(n, i, k) option remember; `if`(i*(i+1)/2 (2*n)!*(b(2*n$2, 0)-`if`(n=0, 0, b(2*n$2, n))):
seq(a(n), n=0..22);
-
b[n_, i_, k_] := b[n, i, k] = If[i (i + 1)/2 < n, 0, If[n == 0, 1, If[i < 2, 0, b[n, i - 1, If[i == k, 0, k]]] + If[i == k, 0, b[n - i, Min[n - i, i - 1], k]/i!]]];
a[n_] := (2n)! (b[2n, 2n, 0] - If[n == 0, 0, b[2n, 2n, n]]);
a /@ Range[0, 22] (* Jean-François Alcover, May 02 2020, after Maple *)
Showing 1-3 of 3 results.
Comments