A326493
Sum of multinomials M(n-k; p_1-1, ..., p_k-1), where p = (p_1, ..., p_k) ranges over all partitions of n into distinct parts (k is a partition length).
Original entry on oeis.org
1, 1, 1, 2, 2, 5, 9, 21, 38, 146, 322, 902, 3106, 8406, 35865, 123321, 393691, 1442688, 7310744, 23471306, 129918661, 500183094, 2400722981, 9592382321, 47764284769, 280267554944, 1247781159201, 7620923955225, 36278364107926, 189688942325418, 1124492015730891
Offset: 0
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with(combinat):
a:= n-> add(multinomial(n-nops(p), map(x-> x-1, p)[], 0),
p=select(l-> nops(l)=nops({l[]}), partition(n))):
seq(a(n), n=0..30);
# second Maple program:
b:= proc(n, i, p) option remember; `if`(i*(i+1)/2 b(n$3):
seq(a(n), n=0..31);
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b[n_, i_, p_] := b[n, i, p] = If[i(i+1)/2 < n, 0, If[n==0, p!, b[n, i-1, p] + b[n-i, Min[n-i, i-1], p-1]/(i-1)!]];
a[n_] := b[n, n, n];
a /@ Range[0, 31] (* Jean-François Alcover, Dec 09 2020, after Alois P. Heinz *)
A327711
Sum of multinomials M(n-k; p_1-1, ..., p_k-1), where p = (p_1, ..., p_k) ranges over all partitions of n (k is a partition length).
Original entry on oeis.org
1, 1, 2, 3, 6, 10, 27, 55, 171, 475, 1555, 4915, 20023, 68243, 288024, 1213828, 5435935, 23966970, 121432923, 578757824, 3130381590, 16427772974, 91877826663, 519546134163, 3199523135912, 18868494152257, 120274458082095, 772954621249540, 5219747666882153
Offset: 0
-
with(combinat):
a:= n-> add(multinomial(n-nops(p), map(
x-> x-1, p)[], 0), p=partition(n)):
seq(a(n), n=0..28);
# second Maple program:
b:= proc(n, i, p) option remember; `if`(n=0, p!, `if`(i<2, 0,
b(n, i-1, p)) +b(n-i, min(n-i, i), p-1)/(i-1)!)
end:
a:= n-> b(n$3):
seq(a(n), n=0..28);
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b[n_, i_, p_] := b[n, i, p] = If[n == 0, p!, If[i < 2, 0, b[n, i - 1, p]] + b[n - i, Min[n - i, i], p - 1]/(i - 1)!];
a[n_] := b[n, n, n];
a /@ Range[0, 28] (* Jean-François Alcover, May 01 2020, from 2nd Maple program *)
A364281
Number of permutations of [n] with distinct cycle lengths such that each cycle contains exactly one cycle length as an element.
Original entry on oeis.org
1, 1, 1, 4, 10, 48, 252, 1584, 10800, 93600, 823680, 8588160, 93381120, 1158312960, 14805504000, 215028172800, 3159494553600, 51973589606400, 873152856576000, 16058241239040000, 300754643245056000, 6159522883497984000, 127439374149255168000
Offset: 0
a(3) = 4: (123), (132), (13)(2), (1)(23).
a(4) = 10: (1234), (1243), (1324), (1342), (1423), (1432), (124)(3),
(142)(3), (1)(234), (1)(243).
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a:= proc(m) option remember; local b; b:=
proc(n, i, p) option remember; `if`(i*(i+1)/2
-
a[m_] := a[m] = Module[{b}, b[n_, i_, p_] := b[n, i, p] = If[i(i+1)/2 < n, 0, If[n == 0, p!*(m - p)!, b[n, i - 1, p] + b[n - i, Min[n - i, i - 1], p - 1]]]; b[m, m, m]];
Table[a[n], {n, 0, 24}] (* Jean-François Alcover, Oct 21 2023, after Alois P. Heinz *)
Showing 1-3 of 3 results.
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