A365821 -id:A365821 - cp's OEIS frontend

A124327 Triangle read by rows: T(n,k) is the number of partitions of the set {1,2,...,n} such that the sum of the least entries of the blocks is k (1<=k<=n*(n+1)/2).

1, 1, 0, 1, 1, 0, 2, 1, 0, 1, 1, 0, 4, 2, 1, 3, 2, 1, 0, 1, 1, 0, 8, 4, 2, 10, 6, 7, 2, 5, 3, 2, 1, 0, 1, 1, 0, 16, 8, 4, 29, 19, 21, 14, 23, 14, 18, 10, 7, 7, 5, 3, 2, 1, 0, 1, 1, 0, 32, 16, 8, 85, 56, 64, 42, 101, 62, 75, 69, 47, 54, 38, 38, 24, 23, 10, 13, 7, 5, 3, 2, 1, 0, 1, 1, 0, 64, 32, 16

Offset: 1

Emeric Deutsch, Oct 31 2006

Row n has n(n+1)/2 terms. Row sums yield the Bell numbers (A000110). T(n,1)=1; T(n,2)=0; T(n,3)=2^(n-2). for n>=2; T(n,4)=2^(n-3) for n>=3; T(n,5)=2^(n-4) for n>=4.

			T(4,7) = 2 because we have 13|2|4 and 1|23|4.
Triangle starts:
  1;
  1, 0,  1;
  1, 0,  2, 1, 0,  1;
  1, 0,  4, 2, 1,  3,  2,  1,  0,  1;
  1, 0,  8, 4, 2, 10,  6,  7,  2,  5,  3,  2,  1, 0, 1;
  1, 0, 16, 8, 4, 29, 19, 21, 14, 23, 14, 18, 10, 7, 7, 5, 3, 2, 1, 0, 1;
  ...

Alois P. Heinz, Rows n = 1..40, flattened

Cf. A000110, A000217, A124325, A367955.
Antidiagonal sums give A365821.
Row maxima give A365903.
T(n,n) gives A368204.

Q[1]:=t*s: for n from 2 to 8 do Q[n]:=expand(s*diff(Q[n-1],s)+t^n*s*Q[n-1]) od: for n from 1 to 8 do P[n]:=sort(subs(s=1,Q[n])) od: for n from 1 to 8 do seq(coeff(P[n],t,k),k=1..n*(n+1)/2) od; # yields sequence in triangular form

Q[1, t_, s_] := t s;
Q[n_, t_, s_] := Q[n, t, s] = s D[Q[n-1, t, s], s] + s t^n Q[n-1, t, s] // Expand;
P[n_, t_] := Q[n, t, s] /. s -> 1;
T[n_] := Rest@CoefficientList[P[n, t], t];
Table[T[n], {n, 1, 8}] // Flatten (* Jean-François Alcover, Jun 10 2024 *)

The generating polynomial of row n is P(n,t)=Q(n,t,1), where Q(n,t,s)=s*dQ(n-1,t,s)/ds + st^n*Q(n-1,t,s); Q(1,t,s)=ts.

Sum_{k=1..n*(n+1)/2} k * T(n,k) = A124325(n+1). - Alois P. Heinz, Dec 05 2023

A368102 Total number of partitions of [n-s] whose block maxima sum to s, summed over all s.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 1, 1, 3, 2, 3, 6, 5, 9, 14, 13, 20, 40, 31, 56, 86, 105, 127, 227, 244, 394, 520, 665, 911, 1536, 1565, 2449, 3507, 4719, 5931, 9061, 11151, 16911, 21774, 29798, 39804, 60411, 71865, 104399, 144999, 197907, 253430, 370044, 478764, 694807

Offset: 0

Alois P. Heinz, Dec 11 2023

nonn

			a(11) = 6: 123|4, 124|3, 13|24, 14|23, 1|2|34, 1|2345.

Alois P. Heinz, Table of n, a(n) for n = 0..5000
Wikipedia, Partition of a set

Antidiagonal sums of A367955.

Cf. A365821.

b:= proc(n, m) option remember; `if`(n=0, 1,
      b(n-1, m)*m + expand(x^n*b(n-1, m+1)))
    end:
a:= n-> add(coeff(b(n-k, 0), x, k), k=ceil(n/2)..n):
seq(a(n), n=0..80);
# second Maple program:
b:= proc(n, i, t) option remember; `if`(i*(i+1)/2 add(b(k, n-k, 0), k=ceil(n/2)..n):
seq(a(n), n=0..80);

b[n_, i_, t_] := b[n, i, t] = If[i*(i + 1)/2 < n, 0, If[n == 0, t^i, If[t == 0, 0, t*b[n, i - 1, t]] + (t + 1)^Max[0, 2*i - n - 1]*b[n - i, Min[n - i, i - 1], t + 1]]];
a[n_] := If[n == 0, 1, Sum[b[k, n - k, 0], {k, Ceiling[n/2], n}]];
Table[a[n], {n, 0, 80}] (* Jean-François Alcover, Oct 03 2024, after Alois P. Heinz *)

cp's OEIS Frontend

A124327 Triangle read by rows: T(n,k) is the number of partitions of the set {1,2,...,n} such that the sum of the least entries of the blocks is k (1<=k<=n*(n+1)/2).

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