A271737
Number of set partitions of [n] with maximal block length multiplicity equal to eight.
Original entry on oeis.org
1, 0, 45, 165, 1980, 14157, 123123, 1042470, 11229075, 117721175, 1085614101, 11354532696, 132028149240, 1440550986525, 15693895739115, 183700174158435, 2200557929261230, 26295830857171150, 323510486572841425, 4085513198322259275, 52716487743732737925
Offset: 8
-
with(combinat):
b:= proc(n, i, k) option remember; `if`(n=0, 1,
`if`(i<1, 0, add(multinomial(n, n-i*j, i$j)
*b(n-i*j, i-1, k)/j!, j=0..min(k, n/i))))
end:
a:= n-> b(n$2, 8)-b(n$2, 7):
seq(a(n), n=8..30);
-
multinomial[n_, k_List] := n!/Times @@ (k!);
b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[multinomial[n, Join[{n - i*j}, Table[i, j]]]*b[n - i*j, i - 1, k]/j!, {j, 0, Min[k, n/i] }]]];
a[n_] := b[n, n, 8] - b[n, n, 7];
Table[a[n], {n, 8, 30}] (* Jean-François Alcover, May 08 2018, after Alois P. Heinz *)
A271738
Number of set partitions of [n] with maximal block length multiplicity equal to nine.
Original entry on oeis.org
1, 0, 55, 220, 2860, 22022, 205205, 1853280, 17381650, 200982925, 2291851991, 23049864630, 262234646310, 3319690300850, 39333605649855, 464026283957060, 5880153732068000, 75836425964702975, 973764622911909400, 12796285021434965050, 173456578124336807300
Offset: 9
-
with(combinat):
b:= proc(n, i, k) option remember; `if`(n=0, 1,
`if`(i<1, 0, add(multinomial(n, n-i*j, i$j)
*b(n-i*j, i-1, k)/j!, j=0..min(k, n/i))))
end:
a:= n-> b(n$2, 9)-b(n$2, 8):
seq(a(n), n=9..30);
-
multinomial[n_, k_List] := n!/Times @@ (k!);
b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[multinomial[n, Join[{n - i*j}, Table[i, j]]]*b[n - i*j, i - 1, k]/j!, {j, 0, Min[k, n/i] }]]];
a[n_] := b[n, n, 9] - b[n, n, 8];
Table[a[n], {n, 9, 30}] (* Jean-François Alcover, May 08 2018, after Alois P. Heinz *)
A271739
Number of set partitions of [n] with maximal block length multiplicity equal to ten.
Original entry on oeis.org
1, 0, 66, 286, 4004, 33033, 328328, 3150576, 31286970, 316394650, 3928974907, 48404715723, 526502083107, 6475762500130, 88834932638892, 1136875206056150, 14448572171583550, 197345257083676845, 2738327374576989195, 37603158111513714720, 528367079280330690400
Offset: 10
-
with(combinat):
b:= proc(n, i, k) option remember; `if`(n=0, 1,
`if`(i<1, 0, add(multinomial(n, n-i*j, i$j)
*b(n-i*j, i-1, k)/j!, j=0..min(k, n/i))))
end:
a:= n-> b(n$2, 10)-b(n$2, 9):
seq(a(n), n=10..30);
-
multinomial[n_, k_List] := n!/Times @@ (k!);
b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[multinomial[n, Join[{n - i*j}, Table[i, j]]]*b[n - i*j, i - 1, k]/j!, {j, 0, Min[k, n/i] }]]];
a[n_] := b[n, n, 10] - b[n, n, 9];
Table[a[n], {n, 10, 30}] (* Jean-François Alcover, May 08 2018, after Alois P. Heinz *)
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