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.

Showing 1-6 of 6 results.

A372762 Number T(n,k) of partitions of [n] having exactly k blocks of minimal size; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 4, 0, 1, 0, 5, 9, 0, 1, 0, 31, 10, 10, 0, 1, 0, 82, 70, 35, 15, 0, 1, 0, 344, 336, 140, 35, 21, 0, 1, 0, 1661, 1393, 616, 385, 56, 28, 0, 1, 0, 7942, 6210, 4984, 1386, 504, 84, 36, 0, 1, 0, 38721, 41331, 22590, 8610, 3717, 840, 120, 45, 0, 1
Offset: 0

Views

Author

Alois P. Heinz, May 12 2024

Keywords

Examples

			T(5,1) = 31: 12345, 1234|5, 1235|4, 123|45, 1245|3, 124|35, 125|34, 12|345, 12|34|5, 12|35|4, 12|3|45, 1345|2, 134|25, 135|24, 13|245, 13|24|5, 13|25|4, 13|2|45, 145|23, 14|235, 14|23|5, 15|234, 1|2345, 15|23|4, 1|23|45, 14|25|3, 14|2|35, 15|24|3, 1|24|35, 15|2|34, 1|25|34.
T(5,2) = 10: 123|4|5, 124|3|5, 125|3|4, 134|2|5, 135|2|4, 1|234|5, 1|235|4, 145|2|3, 1|245|3, 1|2|345.
T(5,3) = 10: 12|3|4|5, 13|2|4|5, 1|23|4|5, 14|2|3|5, 1|24|3|5, 1|2|34|5, 15|2|3|4, 1|25|3|4, 1|2|35|4, 1|2|3|45.
T(5,4) = 0.
T(5,5) = 1: 1|2|3|4|5.
Triangle T(n,k) begins:
  1;
  0,     1;
  0,     1,     1;
  0,     4,     0,     1;
  0,     5,     9,     0,    1;
  0,    31,    10,    10,    0,    1;
  0,    82,    70,    35,   15,    0,   1;
  0,   344,   336,   140,   35,   21,   0,   1;
  0,  1661,  1393,   616,  385,   56,  28,   0,  1;
  0,  7942,  6210,  4984, 1386,  504,  84,  36,  0, 1;
  0, 38721, 41331, 22590, 8610, 3717, 840, 120, 45, 0, 1;
  ...

Links

Crossrefs

Columns k=0-2 give: A000007, A224219, A372764.
Row sums give A000110.
T(2n,n) gives A271425.
Cf. A372650, A372722.

Programs

  • Maple
    b:= proc(n, m, t) option remember; `if`(n=0, x^t,
      add(binomial(n-1, j-1)*b(n-j, min(j, m),
     `if`(j (p-> seq(coeff(p, x, i), i=0..n))(b(n$2, 0)):
seq(T(n), n=0..12);

Formula

Sum_{k=0..n} k * T(n,k) = A372650(n).

A224244 Number of set partitions of {1,2,...,n} such that the size of the smallest block is unique and it contains the element 1.

Original entry on oeis.org

1, 1, 2, 2, 9, 17, 63, 261, 1088, 4374, 24583, 133861, 740303, 4514824, 29945555, 205127474, 1464586617, 10971233035, 86410874373, 708423380237, 6026435657580, 53117555943951, 485246803230148, 4589013046619689, 44819208415713035, 451184268041122808
Offset: 1

Views

Author

Geoffrey Critzer, Apr 01 2013

Keywords

Examples

			a(5) = 9 because we have: {{1,2,3,4,5}}, {{1},{2,3,4,5}}, {{1,2},{3,4,5}}, {{1,3},{2,4,5}}, {{1,5},{2,3,4}}, {{1,4},{2,3,5}}, {{1},{2,3},{4,5}}, {{1},{2,5},{3,4}}, {{1},{2,4},{3,5}}.

Links

Crossrefs

Cf. A224219.

Programs

  • Maple
    b:= proc(n, t) option remember; `if`(n=0, 1, add(
      binomial(n-1, i-1)*b(n-i, `if`(t=1, i+1, t)), i=t..n))
    end:
a:= n-> `if`(n=0, 0, b(n, 1)):
seq(a(n), n=1..30);  # Alois P. Heinz, Jul 07 2016
  • Mathematica
    nn=20;Drop[Range[0,nn]!CoefficientList[Series[Sum[Integrate[x^(k-1)/(k-1)! Exp[Exp[x]-Sum[x^i/i!,{i,0,k}]],x],{k,1,nn}],{x,0,nn}],x],1]
(* Second program: *)
b[n_, t_] := b[n, t] = If[n==0, 1, Sum[Binomial[n-1, i-1]*b[n-i, If[t==1, i + 1, t]], {i, t, n}]]; a[n_] := If[n==0, 0, b[n, 1]]; Table[a[n], {n, 1, 30}] (* Jean-François Alcover, Feb 08 2017, after Alois P. Heinz *)

Formula

E.g.f.: Sum_{k>=1} Integral of x^(k-1)/(k-1)! * exp(exp(x) - Sum_{i=0..k} x^i/i!) dx.

A372721 Number of partitions of [n] having exactly one block of maximal size.

Original entry on oeis.org

0, 1, 1, 4, 11, 36, 132, 596, 2809, 14608, 79448, 461748, 2844052, 18559360, 127712483, 925057295, 7012810967, 55513992168, 457415487326, 3913510354554, 34702368052772, 318406785389976, 3018747693634775, 29537880351353635, 297953826680083794, 3095201088676962296
Offset: 0

Views

Author

Alois P. Heinz, May 11 2024

Keywords

Examples

			a(1) = 1: 1.
a(2) = 1: 12.
a(3) = 4: 123, 12|3, 13|2, 1|23.
a(4) = 11: 1234, 123|4, 124|3, 12|3|4, 134|2, 13|2|4, 1|234, 1|23|4, 14|2|3, 1|24|3, 1|2|34.
a(5) = 36: 12345, 1234|5, 1235|4, 123|45, 123|4|5, 1245|3, 124|35, 124|3|5, 125|34, 12|345, 125|3|4, 12|3|4|5, 1345|2, 134|25, 134|2|5, 135|24, 13|245, 135|2|4, 13|2|4|5, 145|23, 14|235, 15|234, 1|2345, 1|234|5, 1|235|4, 1|23|4|5, 145|2|3, 14|2|3|5, 1|245|3, 1|24|3|5, 1|2|345, 1|2|34|5, 15|2|3|4, 1|25|3|4, 1|2|35|4, 1|2|3|45.

Links

Crossrefs

Column k=1 of A372722.
Cf. A000110, A224219, A372802.

Programs

  • Maple
    b:= proc(n, m, t) option remember; `if`(n=0, `if`(t=1, 1, 0),
      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..25);
  • PARI
    A_x(N) = {my(x='x+O('x^N), f=x+sum(k=2,N, (x^k)/(k!)*exp(sum(j=1,k-1, (x^j)/(j!))))); concat([0],Vec(serlaplace(f)))}
A_x(30) \\ John Tyler Rascoe, Sep 09 2024

Formula

E.g.f: Sum_{k>0} ((x^k)/(k!) * exp(Sum_{j=1..k-1} (x^j)/(j!))). - John Tyler Rascoe, Sep 09 2024

A372802 Number of partitions of [n] having exactly one block of maximal size and one block of minimal size (and any number of non-extremal blocks).

Original entry on oeis.org

0, 1, 1, 4, 5, 16, 82, 169, 1381, 4162, 34346, 109099, 1114610, 5041271, 39441963, 269812729, 1972727781, 14983080612, 126099739072, 989666749503, 8839669627570, 79767000198673, 725399587976669, 6979798715649335, 69812296785011890, 703554021895986941
Offset: 0

Views

Author

Alois P. Heinz, May 13 2024

Keywords

Comments

Minimal block and maximal block are identical if there is only one block.

Examples

			a(1) = 1: 1.
a(2) = 1: 12.
a(3) = 4: 123, 12|3, 13|2, 1|23.
a(4) = 5: 1234, 123|4, 124|3, 134|2, 1|234.
a(5) = 16: 12345, 1234|5, 1235|4, 123|45, 1245|3, 124|35, 125|34, 12|345, 1345|2, 134|25, 135|24, 13|245, 145|23, 14|235, 15|234, 1|2345.
a(8) = 1381: 12345678, 1234567|8, 1234568|7, ..., 1|27|38|456, 18|2|37|456, 1|28|37|456.

Links

Crossrefs

Cf. A000110, A007837, A224219, A372721.

Programs

  • Maple
    b:= proc(n, i) option remember; `if`(n=i, 1, 0)+`if`(i signum(n)+add(binomial(n,i)*b(n-i, i+1), i=1..(n-1)/2):
seq(a(n), n=0..30);

A224245 Number of n-permutations in which there is a unique smallest cycle.

Original entry on oeis.org

1, 1, 5, 14, 89, 474, 3499, 27040, 253161, 2426300, 27596051, 323960856, 4277055925, 59041067344, 898062119655, 14172430400864, 243919993681649, 4347177953716080, 83224487266425811, 1653277176082392040, 34961357216796300381, 763702067489722288136
Offset: 1

Views

Author

Geoffrey Critzer, Apr 01 2013

Keywords

Comments

In other words, if the smallest cycle in the n-permutation has length k then no other cycle in the permutation has length k.

Examples

			a(4) = 14 because we have 14 such permutations of {1,2,3,4} shown in cycle notation: {{1}, {3,4,2}}, {{1}, {4,3,2}}, {{2,3,1}, {4}}, {{2,3,4,1}}, {{2,4,3,1}}, {{2,4,1}, {3}}, {{3,2,1}, {4}}, {{3,4,2,1}}, {{3,4,1}, {2}}, {{3,2,4,1}}, {{4,3,2,1}}, {{4,2,1}, {3}}, {{4,3,1}, {2}}, {{4,2,3,1}}.

Links

Crossrefs

Cf. A224219, A224244.

Programs

  • Maple
    with(combinat):
b:= proc(n, i) option remember;
      `if`(i<1, 0, `if`(n=i, (i-1)!, 0) +add(b(n-i*j, i-1)*
       multinomial(n, n-i*j, i$j)/j!*(i-1)!^j, j=0..(n-1)/i))
    end:
a:= n-> b(n$2):
seq(a(n), n=1..25);  # Alois P. Heinz, Sep 07 2020
  • Mathematica
    nn=20; Drop[Range[0,nn]! CoefficientList[Series[Sum[x^k/k Exp[-Sum[x^i/i, {i,1,k}]]/(1-x), {k,1,nn}], {x,0,nn}], x], 1]

Formula

E.g.f.: Sum_{k>=1} x^k/k * exp(-Sum_{i=1..k}x^i/i)/(1-x).

A224246 The number of n-permutations that have a unique smallest cycle and this cycle contains the element 1.

Original entry on oeis.org

1, 1, 3, 8, 41, 194, 1309, 9022, 79057, 689588, 7462601, 80632826, 1021071193, 13120783948, 192752054377, 2848878770774, 47617784530529, 800500650553472, 14910497765819137, 281133366288649138, 5803224036600349801, 120681837753825004796, 2734647516979262677673, 62424209302423879016558, 1535507329367939907583057
Offset: 1

Views

Author

Geoffrey Critzer, Apr 01 2013

Keywords

Examples

			a(4) = 8 because we have the permutations of {1,2,3,4} in cycle notation:
{{1}, {3,4,2}}, {{1}, {4,3,2}}, {{2,3,4,1}}, {{2,4,3,1}}, {{3,4,2,1}}, {{3,2,4,1}}, {{4,3,2,1}}, {{4,2,3,1}}.

Links

Crossrefs

Cf. A224245, A224244, A224219.

Programs

  • Maple
    b:= proc(n, t) option remember; `if`(n=0, 1, add((i-1)!*
      binomial(n-1, i-1)*b(n-i, `if`(t=1, i+1, t)), i=t..n))
    end:
a:= n-> b(n, 1):
seq(a(n), n=1..30);  # Alois P. Heinz, Sep 07 2020
  • Mathematica
    nn=20; Drop[Range[0,nn]! CoefficientList[Series[Sum[Integrate[x^(k-1) Exp[-Sum[x^i/i,{i,1,k}]]/(1-x),x], {k,1,nn}], {x,0,nn}], x],1]

Formula

E.g.f.: Sum_{k>=1} Integral_((x^(k-1)/(k-1))*exp(-Sum_{i=1..k} x^i/i)/(1-x) dx).
Showing 1-6 of 6 results.

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