A365903 -id:A365903 - 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

A367969 Number of partitions of [n] whose block maxima sum to k, where k is chosen so as to maximize this number.

Original entry on oeis.org

1, 1, 1, 2, 5, 11, 37, 129, 431, 1921, 9544, 43844, 223512, 1407320, 8519457, 52422985, 373424140, 2685768084, 20354852852, 160370778238, 1318493838635, 11239312718146, 98700416575613, 916309760098349, 8735277842452542, 84921152781222758, 860903677319960583

Offset: 0

Alois P. Heinz, Dec 06 2023

nonn

			a(5) = 11 = A367955(5,12) is the largest value in row 5 of A367955 and counts the partitions of [5] having block maxima sum 12: 123|4|5, 124|3|5, 125|3|4, 13|24|5, 13|25|4, 14|23|5, 15|23|4, 14|25|3, 15|24|3, 1|2|34|5, 1|2|35|4.

Alois P. Heinz, Table of n, a(n) for n = 0..200
Wikipedia, Partition of a set

Row maxima of A367955.

Cf. A365441, A365903.

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-> max(coeffs(b(n, 0))):
seq(a(n), n=0..30);
# second Maple program:
b:= proc(n, i, t) option remember; `if`(i*(i+1)/2 max(seq(b(k, n, 0), k=n..n*(n+1)/2)):
seq(a(n), n=0..30);

b[n_, m_] := b[n, m] = If[n == 0, 1, b[n-1, m]*m + Expand[x^n*b[n-1, m+1]]];
a[n_] := Max[CoefficientList[b[n, 0], x]];
Table[a[n], {n, 0, 30}]
(* second program: *)
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, Max[Table[b[k, n, 0], { k, n, n*(n + 1)/2}]]];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Dec 13 2023, 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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