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 *)
A364277
Number of permutations of [n] such that no cycle contains its length as an element.
Original entry on oeis.org
1, 0, 0, 1, 4, 24, 138, 1032, 8160, 75600, 751680, 8436960, 100679040, 1327052160, 18525024000, 280451808000, 4477627123200, 76690072166400, 1377634946688000, 26328977260185600, 525869478021888000, 11092929741653760000, 243781091314016256000, 5628622656645660672000
Offset: 0
a(3) = 1: (13)(2).
a(4) = 4: (124)(3), (142)(3), (13)(2)(4), (14)(2)(3).
A364278
Number of partitions of [n] with distinct block sizes such that no block contains its size as an element.
Original entry on oeis.org
1, 0, 0, 1, 1, 4, 24, 47, 153, 669, 5628, 13554, 61747, 247170, 1539565, 16979571, 53166394, 268393296, 1382097160, 7831424654, 59720804940, 917256305956, 3326800474687, 20441030261195, 112690616749302, 773175024537549, 5164903931159843, 52976603588044961
Offset: 0
a(0) = 1: () the empty partition.
a(3) = 1: 13|2.
a(4) = 1: 124|3.
a(5) = 4: 1235|4, 124|35, 125|34, 13|245.
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 1-4 of 4 results.
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