A097979 -id:A097979 - cp's OEIS frontend

A377824 Sum of the positions of minimum parts in all compositions of n.

0, 1, 4, 10, 29, 70, 181, 435, 1046, 2470, 5762, 13283, 30371, 68847, 154935, 346433, 770154, 1703152, 3748574, 8214805, 17931172, 38997819, 84531066, 182661514, 393578129, 845777569, 1813017039, 3877390908, 8274351482, 17621535902, 37456091552, 79472869966

Offset: 0

John Tyler Rascoe, Nov 08 2024

			The composition of 7, (1,2,1,1,2) has minimum parts at positions 1, 3, and 4; so it contributes 8 to a(7) = 435.

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A001792, A010054, A011782, A097976, A097979, A377823.

b:= proc(n, i, p) option remember; `if`(i<1, 0,
      `if`(irem(n, i)=0, (j-> (p+j)!/j!*(p+j+1)/2*j)(n/i), 0)+
      add(b(n-i*j, i-1, p+j)/j!, j=0..(n-1)/i))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..31);  # Alois P. Heinz, Nov 12 2024

b[n_, i_, p_] := b[n, i, p] = If[i < 1, 0, If[Mod[n, i] == 0, Function[j, (p + j)!/j!*(p + j + 1)/2*j][n/i], 0] + Sum[b[n - i*j, i - 1, p + j]/j!, {j, 0, (n - 1)/i}]];
a[n_] := b[n, n, 0];
Table[a[n], {n, 0, 31}] (* Jean-François Alcover, Apr 19 2025, after Alois P. Heinz *)

A_xy(N) = {my(x='x+O('x^N), h = sum(m=1,N, sum(i=1,N, ((y^i)*x^m)*((x^(m+1))/(1-x))^(i-1)*(sum(j=0,N-m-i, prod(u=1,j, (x^(m+1))/(1-x)+(y^(u+i))*x^m)))))); h}
P_xy(N) = Pol(A_xy(N), {x})
A_x(N) = {my(px = deriv(P_xy(N),y), y=1); Vecrev(eval(px))}
A_x(20)

G.f.: A(x) = d/dy A(x,y)|{y = 1}, where A(x,y) = Sum{m>0} (Sum_{i>0} (x^m * y^i * (x^(m+1)/(1-x))^(i-1) * (Sum_{j>=0} (Product_{u=1..j} (x^(m+1)/(1-x) + x^m * y^(u+i)) ) ) ) ).

Conjecture: a(n) ~ n^2 * 2^(n-5). - Vaclav Kotesovec, Apr 19 2025

A386497 Number of sets of lists of [n] such that one list is the largest.

Original entry on oeis.org

1, 1, 2, 12, 60, 440, 3390, 33852, 338072, 4116240, 51776730, 736751180, 11075784852, 183142075272, 3157190863190, 59336602681020, 1164223828582320, 24348331444705952, 533422896546272562, 12365952739192923660, 298208300418298756460, 7570420981014167756760

Offset: 0

John Tyler Rascoe, Jul 23 2025

Here sets of lists are set partitions of [n] such that the elements within each block are ordered but the blocks themselves are unordered.

			a(3) = 12 counts: {(1),(2,3)}, {(1),(3,2)}, {(1,2),(3)}, {(1,3),(2)}, {(2),(3,1)}, {(2,1),(3)}, {(1,2,3)}, {(1,3,2)}, {(2,1,3)}, {(2,3,1)}, {(3,1,2)}, {(3,2,1)}.

Alois P. Heinz, Table of n, a(n) for n = 0..444

Cf. A000262, A088009, A088311, A088312, A088313, A097979, A113775, A351884, A386474.

b:= proc(n, m, t) option remember; `if`(n=0, t, add(b(n-j, max(m, j),
      `if`(j>m, 1, `if`(j=m, 0, t)))*(n-1)!*j/(n-j)!, j=1..n))
    end:
a:= n-> b(n, 0, 1):
seq(a(n), n=0..21);  # Alois P. Heinz, Jul 23 2025

With[{m = 21}, CoefficientList[Series[1 + Sum[x^j*Exp[(x - x^j)/(1 - x)], {j, 1, m}], {x, 0, m}], x] * Range[0, m]!] (* Amiram Eldar, Jul 24 2025 *)

B_x(N) = {my(x='x+O('x^(N+1))); Vec(serlaplace(1+sum(j=1,N, x^j*exp((x-x^j)/(1-x)))))}

E.g.f.: 1 + Sum_{j>0} x^j * exp((x - x^j)/(1 - x)).

cp's OEIS Frontend

A377824 Sum of the positions of minimum parts in all compositions of n.

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