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 1-3 of 3 results.

A245980 Number A(n,k) of pairs of endofunctions f, g on [n] satisfying g^k(f(i)) = f(i) for all i in [n]; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 16, 1, 1, 6, 729, 1, 1, 10, 87, 65536, 1, 1, 6, 213, 2200, 9765625, 1, 1, 10, 141, 8056, 84245, 2176782336, 1, 1, 6, 213, 6184, 465945, 4492656, 678223072849, 1, 1, 10, 87, 9592, 387545, 37823616, 315937195, 281474976710656
Offset: 0

Views

Author

Alois P. Heinz, Aug 08 2014

Keywords

Examples

			Square array A(n,k) begins:
0 :        1,     1,      1,      1,      1,      1, ...
1 :        1,     1,      1,      1,      1,      1, ...
2 :       16,     6,     10,      6,     10,      6, ...
3 :      729,    87,    213,    141,    213,     87, ...
4 :    65536,  2200,   8056,   6184,   9592,   2200, ...
5 :  9765625, 84245, 465945, 387545, 682545, 159245, ...

Links

Crossrefs

Columns k=0-10 give: A062206, A239750, A239771, A241015, A245981, A245982, A245983, A245984, A245985, A245986, A245987.
Main diagonal gives A245988.
Cf. A245910.

Programs

  • Maple
    with(numtheory): with(combinat): M:=multinomial:
b:= proc(n, k, p) local l, g; l, g:= sort([divisors(p)[]]),
      proc(k, m, i, t) option remember; local d, j; d:= l[i];
        `if`(i=1, n^m, add(M(k, k-(d-t)*j, (d-t)$j)/j!*
         (d-1)!^j *M(m, m-t*j, t$j) *g(k-(d-t)*j, m-t*j,
        `if`(d-t=1, [i-1, 0], [i, t+1])[]), j=0..min(k/(d-t),
        `if`(t=0, [][], m/t))))
      end; g(k, n-k, nops(l), 0)
    end:
A:= (n, k)-> `if`(k=0, n^(2*n), add(b(n, j, k)*
             stirling2(n, j)*binomial(n, j)*j!, j=0..n)):
seq(seq(A(n, d-n), n=0..d), d=0..12);
  • Mathematica
    multinomial[n_, k_List] := n!/Times @@ (k!); M = multinomial;
b[n_, k0_, p_] := Module[{l, g}, l = Sort[Divisors[p]]; g[k_, m_, i_, t_] := g[k, m, i, t] = Module[{d, j}, d = l[[i]]; If[i == 1, n^m, Sum[M[k, Join[{k - (d-t)*j}, Array[(d - t)&, j]]]/ j!*(d-1)!^j * M[m, Join[{m - t*j}, Array[t&, j]]]*If[d-t == 1, g[k - (d - t)*j, m - t*j, i-1, 0], g[k - (d-t)*j, m - t*j, i, t+1]], {j, 0, Min[k/(d-t), If[t == 0, Infinity, m/t]]}]]]; g[k0, n-k0, Length[l], 0]];
A[n_, k_] := If[k == 0, n^(2*n), Sum[b[n, j, k]*StirlingS2[n, j]* Binomial[n, j]*j!, {j, 0, n}]]; A[0, ] = 1; A[1, ] = 1;
Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 12}] // Flatten (* Jean-François Alcover, Jan 27 2015, after Alois P. Heinz *)

A241015 Number of pairs of endofunctions f, g on [n] satisfying g(g(g(f(i)))) = f(i) for all i in [n].

Original entry on oeis.org

1, 1, 6, 141, 6184, 387545, 33404256, 3891981205, 592320594048, 113184611671473, 26327424526220800, 7302855260707822541, 2381136881374877847552, 901709366369630531857417, 392234247731566637785780224, 194028806625479344354551301125
Offset: 0

Views

Author

Alois P. Heinz, Aug 07 2014

Keywords

Links

Crossrefs

Cf. A048993, A239750, A239771, A245958.
Column k=3 of A245980.

Programs

  • Maple
    with(combinat): M:=multinomial:
b:= proc(n, k) local l, g; l, g:= [1, 3],
      proc(k, m, i, t) option remember; local d, j; d:= l[i];
        `if`(i=1, n^m, add(M(k, k-(d-t)*j, (d-t)$j)/j!*
         (d-1)!^j *M(m, m-t*j, t$j) *g(k-(d-t)*j, m-t*j,
        `if`(d-t=1, [i-1, 0], [i, t+1])[]), j=0..min(k/(d-t),
        `if`(t=0, [][], m/t))))
      end; g(k, n-k, nops(l), 0)
    end:
a:= n-> add(b(n, j)*stirling2(n, j)*binomial(n, j)*j!, j=0..n):
seq(a(n), n=0..20);
  • Mathematica
    multinomial[n_, k_] := n!/Times @@ (k!); M = multinomial; b[n_, k0_] := Module[{l, g}, l = {1, 3}; g[k_, m_, i_, t_] := g[k, m, i, t] = Module[{d, j}, d = l[[i]]; If[i==1, n^m, Sum[M[k, Join[{k-(d-t)*j}, Array[d-t&, j]]]/j!*(d-1)!^j *M[m, Join[{m-t*j}, Array[t&, j]]]*g[k-(d-t)*j, m-t*j, Sequence @@ If[d-t==1, {i-1, 0}, {i, t+1}]], {j, 0, Min[k/(d-t), If[t==0, Infinity, m/t]]}]]]; g[k0, n-k0, Length[l], 0]]; a[0] = 1; a[n_] := Sum[b[n, j]*StirlingS2[n, j]*Binomial[n, j]*j!, {j, 0, n}]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Mar 13 2017, translated from Maple *)

Formula

a(n) = Sum_{k=0..n} C(n,k) * A048993(n,k) * k! * A245958(n,k).

A245348 Number T(n,k) of endofunctions f on [n] that are self-inverse on [k]; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 1, 1, 4, 3, 2, 27, 15, 8, 4, 256, 112, 50, 22, 10, 3125, 1125, 430, 166, 66, 26, 46656, 14256, 4752, 1626, 576, 206, 76, 823543, 218491, 64484, 19768, 6310, 2054, 688, 232, 16777216, 3932160, 1040384, 288512, 83736, 24952, 7660, 2388, 764
Offset: 0

Views

Author

Alois P. Heinz, Jul 18 2014

Keywords

Comments

T(n,k) counts endofunctions f:{1,...,n}-> {1,...,n} with f(f(i))=i for all i in {1,...,k}.

Examples

			T(3,1) = 15: (1,1,1), (2,1,1), (3,1,1), (1,2,1), (3,2,1), (1,3,1), (3,3,1), (1,1,2), (2,1,2), (1,2,2), (1,3,2), (1,1,3), (2,1,3), (1,2,3), (1,3,3).
T(3,2) = 8: (2,1,1), (1,2,1), (3,2,1), (2,1,2), (1,2,2), (1,3,2), (2,1,3), (1,2,3).
T(3,3) = 4: (3,2,1), (1,3,2), (2,1,3), (1,2,3).
Triangle T(n,k) begins:
0 :       1;
1 :       1,      1;
2 :       4,      3,     2;
3 :      27,     15,     8,     4;
4 :     256,    112,    50,    22,   10;
5 :    3125,   1125,   430,   166,   66,   26;
6 :   46656,  14256,  4752,  1626,  576,  206,  76;
7 :  823543, 218491, 64484, 19768, 6310, 2054, 688, 232;
     ...

Links

Crossrefs

Columns k=0-1 give: A000312, A089945(n-1) for n>0.
Main diagonal gives A000085.
T(2n,n) gives A245141.
Cf. A239771, A245692.

Programs

  • Maple
    g:= proc(n) g(n):= `if`(n<2, 1, g(n-1)+(n-1)*g(n-2)) end:
T:= (n, k)-> add(binomial(n-k, i)*binomial(k, i)*i!*
             g(k-i)*n^(n-k-i), i=0..min(k, n-k)):
seq(seq(T(n,k), k=0..n), n=0..10);
  • Mathematica
    g[n_] := g[n] = If[n<2, 1, g[n-1] + (n-1)*g[n-2]]; T[0, 0] = 1; T[n_, k_] := Sum[Binomial[n-k, i]*Binomial[k, i]*i!*g[k-i]*n^(n-k-i), {i, 0, Min[k, n-k]}]; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Feb 19 2017, translated from Maple *)

Formula

T(n,k) = Sum_{i=0..min(k,n-k)} C(n-k,i)*C(k,i)*i!*A000085(k-i)*n^(n-k-i).
Showing 1-3 of 3 results.

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