A224219
Number of set partitions of {1,2,...,n} such that the size of the smallest block is unique.
Original entry on oeis.org
1, 1, 4, 5, 31, 82, 344, 1661, 7942, 38721, 228680, 1377026, 8529756, 56756260, 402300799, 2960135917, 22692746719, 181667760724, 1516381486766, 13135566948285, 117868982320877, 1093961278908818, 10492653292100919, 103880022098900234, 1059925027073166856
Offset: 1
a(4) = 5 because we have: {{1,2,3,4}}, {{1},{2,3,4}}, {{1,3,4},{2}}, {{1,2,3},{4}}, {{1,2,4},{3}}.
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with(combinat):
b:= proc(n, i) option remember;
`if`(i<1, 0, `if`(n=i, 1, 0)+add(b(n-i*j, i-1)*
multinomial(n, n-i*j, i$j)/j!, j=0..(n-1)/i))
end:
a:= n-> b(n$2):
seq(a(n), n=1..25); # Alois P. Heinz, Jul 07 2016
-
nn=25;Drop[Range[0,nn]!CoefficientList[Series[Sum[x^k/k!Exp[Exp[x]-Sum[x^i/i!,{i,0,k}]],{k,1,nn}],{x,0,nn}],x],1]
(* Second program: *)
multinomial[n_, k_List] := n!/Times @@ (k!); b[n_, i_] := b[n, i] = If[i<1, 0, If[n==i, 1, 0] + Sum[b[n-i*j, i-1]*multinomial[n, Prepend[Array[i&, j], n-i*j]]/j!, {j, 0, (n-1)/i}]]; a[n_] := b[n, n]; Table[a[n], {n, 1, 25}] (* Jean-François Alcover, Feb 03 2017, after Alois P. Heinz *)
A372722
Number T(n,k) of partitions of [n] having exactly k blocks of maximal size; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
Original entry on oeis.org
1, 0, 1, 0, 1, 1, 0, 4, 0, 1, 0, 11, 3, 0, 1, 0, 36, 15, 0, 0, 1, 0, 132, 55, 15, 0, 0, 1, 0, 596, 175, 105, 0, 0, 0, 1, 0, 2809, 805, 420, 105, 0, 0, 0, 1, 0, 14608, 4053, 1540, 945, 0, 0, 0, 0, 1, 0, 79448, 24906, 5950, 4725, 945, 0, 0, 0, 0, 1, 0, 461748, 151371, 37730, 17325, 10395, 0, 0, 0, 0, 0, 1
Offset: 0
T(4,1) = 11: 1234, 123|4, 124|3, 12|3|4, 134|2, 13|2|4, 1|234, 1|23|4, 14|2|3, 1|24|3, 1|2|34.
T(4,2) = 3: 12|34, 13|24, 14|23.
T(4,3) = 0.
T(4,4) = 1: 1|2|3|4.
Triangle T(n,k) begins:
1;
0, 1;
0, 1, 1;
0, 4, 0, 1;
0, 11, 3, 0, 1;
0, 36, 15, 0, 0, 1;
0, 132, 55, 15, 0, 0, 1;
0, 596, 175, 105, 0, 0, 0, 1;
0, 2809, 805, 420, 105, 0, 0, 0, 1;
0, 14608, 4053, 1540, 945, 0, 0, 0, 0, 1;
0, 79448, 24906, 5950, 4725, 945, 0, 0, 0, 0, 1;
...
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b:= proc(n, m, t) option remember; `if`(n=0, x^t,
add(binomial(n-1, j-1)*b(n-j, max(j, m),
`if`(j>m, 1, `if`(j=m, t+1, t))), j=1..n))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n, 0$2)):
seq(T(n), n=0..12);
A372802
Number of partitions of [n] having exactly one block of maximal size and one block of minimal size (and any number of non-extremal blocks).
Original entry on oeis.org
0, 1, 1, 4, 5, 16, 82, 169, 1381, 4162, 34346, 109099, 1114610, 5041271, 39441963, 269812729, 1972727781, 14983080612, 126099739072, 989666749503, 8839669627570, 79767000198673, 725399587976669, 6979798715649335, 69812296785011890, 703554021895986941
Offset: 0
a(1) = 1: 1.
a(2) = 1: 12.
a(3) = 4: 123, 12|3, 13|2, 1|23.
a(4) = 5: 1234, 123|4, 124|3, 134|2, 1|234.
a(5) = 16: 12345, 1234|5, 1235|4, 123|45, 1245|3, 124|35, 125|34, 12|345, 1345|2, 134|25, 135|24, 13|245, 145|23, 14|235, 15|234, 1|2345.
a(8) = 1381: 12345678, 1234567|8, 1234568|7, ..., 1|27|38|456, 18|2|37|456, 1|28|37|456.
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b:= proc(n, i) option remember; `if`(n=i, 1, 0)+`if`(i signum(n)+add(binomial(n,i)*b(n-i, i+1), i=1..(n-1)/2):
seq(a(n), n=0..30);
Showing 1-3 of 3 results.
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