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 *)
A364279
Number of permutations of [n] with distinct cycle lengths such that no cycle contains its length as an element.
Original entry on oeis.org
1, 0, 0, 1, 2, 12, 86, 546, 4284, 39588, 416988, 4378848, 54297504, 695592000, 9840307680, 149031686880, 2387863575360, 40338090711360, 736126007279040, 13938942123429120, 279358800902737920, 5894877845100625920, 129943826126987765760, 2985640822908446976000
Offset: 0
a(3) = 1: (13)(2).
a(4) = 2: (124)(3), (142)(3).
a(5) = 12: (1235)(4), (1253)(4), (1325)(4), (1352)(4), (1523)(4), (1532)(4), (124)(35), (142)(35), (125)(34), (152)(34), (13)(245), (13)(254).
A364282
Number of partitions of [n] with distinct block sizes such that each block contains exactly one block size different from its own as an element.
Original entry on oeis.org
1, 0, 0, 1, 1, 4, 11, 24, 52, 226, 969, 2281, 8960, 29898, 193202, 1075509, 3346852, 14280775, 75858992, 332978617, 2839114204, 19507400962, 75453432614, 383685116089, 2030801987312, 14025840725149, 77948290561659, 884660446815877, 7273497958681824
Offset: 0
a(3) = 1: 13|2.
a(4) = 1: 124|3.
a(5) = 4: 1235|4, 124|35, 125|34, 13|245.
a(6) = 11: 12346|5, 1235|46, 1236|45, 1256|34, 14|2356, 145|2|36, 14|256|3, 146|2|35, 15|246|3, 16|245|3, 156|2|34.
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f:= proc(n) option remember; `if`(n<2, 1-n, (n-1)*(f(n-1)+f(n-2))) end:
a:= proc(m) option remember; local b; b:=
proc(n, i, p) option remember; `if`(i*(i+1)/2
Showing 1-3 of 3 results.
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