A319375
Number T(n,k) of entries in the k-th blocks of all set partitions of [n] when blocks are ordered by decreasing lengths (and increasing smallest elements); triangle T(n,k), n>=1, 1<=k<=n, read by rows.
Original entry on oeis.org
1, 3, 1, 10, 4, 1, 35, 17, 7, 1, 136, 76, 36, 11, 1, 577, 357, 186, 81, 16, 1, 2682, 1737, 1023, 512, 162, 22, 1, 13435, 8997, 5867, 3151, 1345, 295, 29, 1, 72310, 49420, 34744, 20071, 10096, 3145, 499, 37, 1, 414761, 289253, 211888, 133853, 72973, 29503, 6676, 796, 46, 1
Offset: 1
The 5 set partitions of {1,2,3} are:
1 |2 |3
12 |3
13 |2
23 |1
123
so there are 10 elements in the first (largest) blocks, 4 in the second blocks and only 1 in the third blocks.
Triangle T(n,k) begins:
1;
3, 1;
10, 4, 1;
35, 17, 7, 1;
136, 76, 36, 11, 1;
577, 357, 186, 81, 16, 1;
2682, 1737, 1023, 512, 162, 22, 1;
13435, 8997, 5867, 3151, 1345, 295, 29, 1;
72310, 49420, 34744, 20071, 10096, 3145, 499, 37, 1;
...
-
b:= proc(n, l) option remember; `if`(n=0, add(l[-i]*
x^i, i=1..nops(l)), add(binomial(n-1, j-1)*
b(n-j, sort([l[], j])), j=1..n))
end:
T:= n-> (p-> (seq(coeff(p, x, i), i=1..n)))(b(n, [])):
seq(T(n), n=1..12);
# second Maple program:
b:= proc(n, i, t) option remember; `if`(n=0, [1, 0], `if`(i<1, 0,
add((p-> p+`if`(t>0 and t-j<1, [0, p[1]*i], 0))(
combinat[multinomial](n, i$j, n-i*j)/j!*
b(n-i*j, min(n-i*j, i-1), max(0, t-j))), j=0..n/i)))
end:
T:= (n, k)-> b(n$2, k)[2]:
seq(seq(T(n, k), k=1..n), n=1..12); # Alois P. Heinz, Mar 02 2020
-
b[n_, l_] := b[n, l] = If[n == 0, Sum[l[[-i]] x^i, {i, 1, Length[l]}], Sum[ Binomial[n-1, j-1] b[n-j, Sort[Append[l, j]]], {j, 1, n}]];
T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 1, n}]][b[n, {}]];
Array[T, 12] // Flatten (* Jean-François Alcover, Dec 28 2018, after Alois P. Heinz *)
A097147
Total sum of minimum block sizes in all partitions of n-set.
Original entry on oeis.org
1, 3, 7, 21, 66, 258, 1079, 4987, 25195, 136723, 789438, 4863268, 31693715, 217331845, 1564583770, 11795630861, 92833623206, 760811482322, 6479991883525, 57256139503047, 523919025038279, 4956976879724565, 48424420955966635, 487810283307069696
Offset: 1
-
g:= proc(n, i, p) option remember; `if`(n=0, (i+1)*p!,
`if`(i<1, 0, add(g(n-i*j, i-1, p+j*i)/j!/i!^j, j=0..n/i)))
end:
a:= n-> g(n$2, 0):
seq(a(n), n=1..30); # Alois P. Heinz, Mar 06 2015
-
Drop[Apply[Plus,Table[nn=25;Range[0,nn]!CoefficientList[Series[Exp[Sum[ x^i/i!,{i,n,nn}]]-1,{x,0,nn}],x],{n,1,nn}]],1] (* Geoffrey Critzer, Jan 10 2013 *)
g[n_, i_, p_] := g[n, i, p] = If[n == 0, (i+1)*p!, If[i<1, 0,
Sum[g[n-i*j, i-1, p+j*i]/j!/i!^j, {j, 0, n/i}]]];
a[n_] := g[n, n, 0];
Array[a, 30] (* Jean-François Alcover, Aug 24 2021, after Alois P. Heinz *)
A097145
Total sum of minimum list sizes in all sets of lists of n-set, cf. A000262.
Original entry on oeis.org
0, 1, 5, 25, 157, 1101, 9211, 85513, 900033, 10402633, 133059331, 1836961941, 27619253113, 444584808253, 7678546353843, 140944884572521, 2751833492404321, 56691826303303953, 1233793951629951043, 28191548364561422173, 676190806704598883241
Offset: 0
For n=4 we have 73 sets of lists (cf. A000262): (1234) (24 ways), (123)(4) (6*4 ways), (12)(34) (3*4 ways), (12)(3)(4) (6*2 ways), (1)(2)(3)(4) (1 way); so a(n)= 24*4+24*1+12*2+12*1+1*1 = 157.
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b:= proc(n, m) option remember; `if`(n=0, m, add(j!*
b(n-j, min(m, j))*binomial(n-1, j-1), j=1..n))
end:
a:= n-> `if`(n=0, 0, b(n, infinity)):
seq(a(n), n=0..25); # Alois P. Heinz, May 10 2016
-
b[n_, m_] := b[n, m] = If[n==0, m, Sum[j!*b[n-j, Min[m, j]]*Binomial[n-1, j - 1], {j, 1, n}]]; a[n_] := If[n==0, 0, b[n, Infinity]]; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Feb 18 2017, after Alois P. Heinz *)
A097146
Total sum of maximum list sizes in all sets of lists of n-set, cf. A000262.
Original entry on oeis.org
0, 1, 5, 31, 217, 1781, 16501, 172915, 1998641, 25468777, 352751941, 5292123431, 85297925065, 1472161501981, 27039872306357, 527253067633531, 10865963240550241, 236088078855319505, 5390956470528548101, 129102989125943058607, 3234053809095307670201, 84596120521251178630981, 2305894874979300173268085
Offset: 0
For n=4 we have 73 sets of lists (cf. A000262): (1234) (24 ways), (123)(4) (6*4 ways), (12)(34) (3*4 ways), (12)(3)(4) (6*2 ways), (1)(2)(3)(4) (1 way); so a(4)= 24*4+24*3+12*2+12*2+1*1 = 217.
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b:= proc(n, m) option remember; `if`(n=0, m, add(j!*
b(n-j, max(m, j))*binomial(n-1, j-1), j=1..n))
end:
a:= n-> b(n, 0):
seq(a(n), n=0..25); # Alois P. Heinz, May 10 2016
-
b[n_, m_] := b[n, m] = If[n == 0, m, Sum[j! b[n-j, Max[m, j]] Binomial[n-1, j-1], {j, 1, n}]];
a[n_] := b[n, 0];
a /@ Range[0, 25] (* Jean-François Alcover, Nov 05 2020, after Alois P. Heinz *)
-
N=50; x='x+O('x^N);
egf=exp(x/(1-x))*sum(k=1,N, (1-exp(x^k/(x-1))) );
Vec( serlaplace(egf) ) /* show terms */
A372649
Total sum over all partitions of [n] of the number of maximal blocks.
Original entry on oeis.org
0, 1, 3, 7, 21, 71, 293, 1268, 6107, 31123, 170745, 998966, 6212627, 40854360, 283290348, 2059884614, 15667307457, 124266461587, 1025342179759, 8784261413616, 78003593175261, 716854898767936, 6808817431686858, 66754426111124686, 674754718441688851
Offset: 0
a(3) = 7 = 3 + 1 + 1 + 1 + 1: 1|2|3, 1|23, 12|3, 13|2, 123.
a(4) = 21 = 1+1+1+2+1+1+2+1+2+1+1+1+1+1+4: 1234, 123|4, 124|3, 12|34, 12|3|4, 134|2, 13|24, 13|2|4, 14|23, 1|234, 1|23|4, 14|2|3, 1|24|3, 1|2|34, 1|2|3|4.
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b:= proc(n, m, t) option remember; `if`(n=0, 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:
a:= n-> b(n, 0$2):
seq(a(n), n=0..24);
-
b[n_, m_, t_] := b[n, m, t] = If[n == 0, t,
Sum[Binomial[n - 1, j - 1]*b[n - j, Max[j, m],
If[j > m, 1, If[j == m, t + 1, t]]], {j, 1, n}]];
a[n_] := b[n, 0, 0];
Table[a[n], {n, 0, 24}] (* Jean-François Alcover, May 10 2024, after Alois P. Heinz *)
Showing 1-5 of 5 results.