cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A367955 Number T(n,k) of partitions of [n] whose block maxima sum to k, triangle T(n,k), n>=0, n<=k<=n*(n+1)/2, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 5, 2, 3, 1, 1, 1, 2, 5, 10, 7, 7, 11, 3, 4, 1, 1, 1, 2, 5, 10, 23, 15, 23, 25, 37, 18, 14, 19, 4, 5, 1, 1, 1, 2, 5, 10, 23, 47, 39, 49, 81, 84, 129, 74, 78, 70, 87, 33, 23, 29, 5, 6, 1, 1, 1, 2, 5, 10, 23, 47, 103, 81, 129, 172, 261, 304, 431, 299, 325, 376, 317, 424, 196, 183, 144, 165, 52, 34, 41, 6, 7, 1
Offset: 0

Views

Author

Alois P. Heinz, Dec 05 2023

Keywords

Comments

Rows and also columns reversed converge to A365441.
T(n,k) is defined for all n,k >= 0. The triangle contains only the positive terms. T(n,k) = 0 if k < n or k > n*(n+1)/2.

Examples

			T(4,7) = 5: 123|4, 124|3, 13|24, 14|23, 1|2|34.
T(5,9) = 10: 1234|5, 1235|4, 124|35, 125|34, 134|25, 135|24, 14|235, 15|234, 1|23|45, 1|245|3.
T(5,13) = 3: 1|23|4|5, 1|24|3|5, 1|25|3|4.
T(5,14) = 4: 12|3|4|5, 13|2|4|5, 14|2|3|5, 15|2|3|4.
T(5,15) = 1: 1|2|3|4|5.
Triangle T(n,k) begins:
  1;
  .  1;
  .  .  1, 1;
  .  .  .  1, 1, 2, 1;
  .  .  .  .  1, 1, 2, 5, 2,  3,  1;
  .  .  .  .  .  1, 1, 2, 5, 10,  7,  7, 11,  3,  4,  1;
  .  .  .  .  .  .  1, 1, 2,  5, 10, 23, 15, 23, 25, 37, 18, 14, 19, 4, 5, 1;
  ...

Links

Crossrefs

Row sums give A000110.
Column sums give A204856.
Antidiagonal sums give A368102.
T(2n,3n) gives A365441.
T(n,2n) gives A368675.
Row maxima give A367969.
Row n has A000124(n-1) terms (for n>=1).
Cf. A000217, A124327 (the same for block minima), A200660, A278677.

Programs

  • Maple
    b:= proc(n, m) option remember; `if`(n=0, 1,
      b(n-1, m)*m + expand(x^n*b(n-1, m+1)))
    end:
T:= (n, k)-> coeff(b(n, 0), x, k):
seq(seq(T(n, k), k=n..n*(n+1)/2), n=0..10);
# second Maple program:
b:= proc(n, i, t) option remember; `if`(i*(i+1)/2 b(k, n, 0):
seq(seq(T(n, k), k=n..n*(n+1)/2), n=0..10);
  • Mathematica
    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]]];
T[0, 0] = 1; T[n_, k_] := b[k, n, 0];
Table[Table[T[n, k], {k, n, n*(n + 1)/2}], {n, 0, 10}] // Flatten (* Jean-François Alcover, Oct 03 2024, after Alois P. Heinz's second Maple program *)

Formula

Sum_{k=n..n*(n+1)/2} k * T(n,k) = A278677(n-1) for n>=1.
Sum_{k=n..n*(n+1)/2} (k-n) * T(n,k) = A200660(n) for n>=1.
T(n,n) = T(n,n*(n+1)/2) = 1.

A365441 Number of partitions of [2n] whose block maxima sum to 3n.

Original entry on oeis.org

1, 1, 2, 5, 10, 23, 47, 103, 209, 449, 908, 1909, 3864, 8011, 16165, 33244, 66933, 136628, 274876, 558107, 1121160, 2268526, 4552291, 9183569, 18417449, 37073504, 74300048, 149334422, 299134695, 600481001, 1202436958, 2411536369, 4827532935, 9675363921, 19364235775
Offset: 0

Views

Author

Alois P. Heinz, Dec 06 2023

Keywords

Comments

Rows of A367955 and the reversed columns of A367955 converge to this sequence.

Examples

			a(0) = 1: the empty partition.
a(1) = 1: 1|2.
a(2) = 2: 12|34, 134|2.
a(3) = 5: 123|456, 12456|3, 13|2456, 1456|23, 1|2|3456.
a(4) = 10: 1234|5678, 1235678|4, 124|35678, 125678|34, 134|25678, 135678|24, 14|235678, 15678|234, 1|23|45678, 1|245678|3.
a(5) = 23: 12345|6789(10), 12346789(10)|5, 1235|46789(10), 1236789(10)|45, 1245|36789(10), 1246789(10)|35, 125|346789(10), 126789(10)|345, 12|3|456789(10), 1345|26789(10), 1346789(10)|25, 135|246789(10), 136789(10)|245, 13|2|456789(10), 145|236789(10), 146789(10)|235, 15|2346789(10), 16789(10)|2345, 1|234|56789(10), 1|2356789(10)|4, 1456789(10)|2|3, 1|24|356789(10), 1|256789(10)|34.

Links

Crossrefs

Cf. A000110, A367955, A367969, A368204, A368675.

Programs

  • Maple
    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-> coeff(b(2*n, 0), x, 3*n):
seq(a(n), n=0..42);
# second Maple program:
b:= proc(n, i, t) option remember; `if`(i*(i+1)/2 b(3*n, 2*n, 0):
seq(a(n), n=0..42);
  • Mathematica
    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, b[3n, 2n, 0]];
Table[a[n], {n, 0, 42}] (* Jean-François Alcover, Oct 03 2024, after Alois P. Heinz *)

Formula

a(n) = A367955(2n,3n).

A365903 Number of partitions of [n] whose block minima sum to k, where k is chosen so as to maximize this number.

Original entry on oeis.org

1, 1, 1, 2, 4, 10, 29, 101, 367, 1562, 6891, 37871, 197930, 1121634, 6888085, 46190282, 323250987, 2349020516, 17897285514, 142512956148, 1178963284732, 10248806222398, 91421283039658, 847666112839362, 8100455404172267, 79925567946537362, 814508927747776069
Offset: 0

Views

Author

Alois P. Heinz, Dec 14 2023

Keywords

Links

Crossrefs

Row maxima of A124327.
Cf. A367969.

Programs

  • Maple
    b:= proc(n, i, t, m) option remember; `if`(n=0, t^(m-i+1),
     `if`((i+m)*(m+1-i)/2n, 0, `if`(t=0, 0,
      t*b(n, i+1, t, m))+ b(n-i, i+1, t+1, m)))
    end:
a:= n-> max(seq(b(k, 1, 0, n), k=0..n*(n+1)/2)):
seq(a(n), n=0..26);
# second Maple program:
a:= proc(h) option remember; local b; b:=
      proc(n, m) option remember; `if`(n=0, 1,
        b(n-1, m)*m + expand(x^(h-n+1)*b(n-1, m+1)))
      end: forget(b); max(coeffs(b(h, 0)))
    end:
seq(a(n), n=0..26);
  • Mathematica
    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_] := Module[{s}, Q[n, t, s] /. s -> 1];
a[n_] := If[n == 0, 1, Module[{t}, CoefficientList[P[n, t], t] // Max]];
Table[a[n], {n, 0, 26}] (* Jean-François Alcover, Oct 03 2024 *)
Showing 1-3 of 3 results.

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