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 1-5 of 5 results.

A261764 Triangle read by rows: T(n,k) is the number of nilpotent subpermutations on an n-set, each of nilpotency index less than or equal to k.

Original entry on oeis.org

1, 0, 1, 0, 1, 3, 0, 1, 7, 13, 0, 1, 25, 49, 73, 0, 1, 81, 261, 381, 501, 0, 1, 331, 1531, 2611, 3331, 4051, 0, 1, 1303, 9073, 19993, 27553, 32593, 37633, 0, 1, 5937, 63393, 165873, 253233, 313713, 354033, 394353, 0, 1, 26785, 465769, 1436473, 2540233, 3326473, 3870793, 4233673, 4596553
Offset: 0

Views

Author

Samira Stitou, Sep 21 2015

Keywords

Examples

			T(3,2) = 7 because there are 7 nilpotent subpermutations on {1,2,3}, each of nilpotency index less than or equal to 2, namely: empty map, 1-->2, 1-->3, 2-->1, 2-->3, 3-->1, 3-->2.
Triangle starts:
1;
0, 1;
0, 1,   3;
0, 1,   7,   13;
0, 1,  25,   49,   73;
0, 1,  81,  261,  381,  501;
0, 1, 331, 1531, 2611, 3331, 4051;
...

References

  • A. Laradji and A. Umar, On the number of subpermutations with fixed orbit size, Ars Combinatoria, 109 (2013), 447-460.

Links

Crossrefs

Cf. A000262, A157400, A261762, A261763, A261765, A261766, A261767.

Programs

  • Maple
    egf:= k-> exp(add(x^j, j=1..k)):
T:= (n, k)-> n!*coeff(series(egf(k), x, n+1), x, n):
seq(seq(T(n, k), k=0..n), n=0..10);  # Alois P. Heinz, Oct 10 2015
# second Maple program:
T:= proc(n, k) option remember; `if`(n=0, 1, add(
      T(n-j, k)*binomial(n-1, j-1)*j!, j=1..min(n,k)))
    end:
seq(seq(T(n, k), k=0..n), n=0..10);  # Alois P. Heinz, Sep 29 2017
  • Mathematica
    Table[n!*SeriesCoefficient[Exp[x*(x^k-1)/(x-1)], {x, 0, n}], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, May 18 2016 *)

Formula

T(n, k) = T(n-1, k) + 2(n-1)T(n-2, k) + ... + k(n-1) ... (n-k+1)T(n-k, k), with T(n, 1) = 1 and T(n, n+r) = T(n, n) for every nonnegative integer r.
T(n,n) = A000262(n).
E.g.f. of column k: exp(x + x^2 + ... + x^k).
T(n,k) = Sum_{i=0..k} A157400(n,i).

Extensions

More terms from Alois P. Heinz, Oct 10 2015

A261767 Triangle read by rows: T(n,k) is the number of subpermutations of an n-set, whose orbits are each of size at most k with at least one orbit of size exactly k.

Original entry on oeis.org

1, 1, 1, 1, 3, 3, 1, 7, 18, 8, 1, 15, 99, 64, 30, 1, 31, 510, 560, 300, 144, 1, 63, 2745, 4800, 3150, 1728, 840
Offset: 0

Views

Author

Samira Stitou, Sep 21 2015

Keywords

Examples

			T(3, 2) = 18 because there are 18 subpermutations on {1,2,3} whose orbits are each of size at most 2 with at least one orbit of size exactly 2, namely: (1 2 --> 2 1), (1 3 --> 3 1), (2 3 --> 3 2), (123 --> 213), (123 --> 321), (123 --> 132); (1-->2), (1-->3), (2-->1), (2-->3), (3-->1), (3-->2); (13-->23), (12-->32), (23-->13), (32-->33), (23-->21), (13-->12).
Triangle starts:
1;
1, 1;
1, 3, 3;
1, 7, 18, 8;
1, 15, 99, 64, 30;
1, 31, 510, 560, 300, 144;
...

References

  • A. Laradji and A. Umar, On the number of subpermutations with fixed orbit size, Ars Combinatoria, 109 (2013), 447-460.

Crossrefs

Cf. A261762, A261763, A261764, A261765, A261766, A261767.
Row sums give A002720.

Formula

T(n, k) = A261763(n, k) - A261763(n, k-1), T(n, n) = A261766(n) for all n not equal to 1 and T(1, 1) = 1.

A261763 Triangle read by rows: T(n,k) is the number of subpermutations of an n-set whose orbits are each of size at most k.

Original entry on oeis.org

1, 1, 2, 1, 4, 7, 1, 8, 26, 34, 1, 16, 115, 179, 209, 1, 32, 542, 1102, 1402, 1546, 1, 64, 2809, 7609, 10759, 12487, 13327, 1, 128, 15374, 56534, 92234, 113402, 125162, 130922, 1, 256, 89737, 457993, 865393, 1139569, 1304209, 1396369, 1441729
Offset: 0

Views

Author

Samira Stitou, Sep 21 2015

Keywords

Examples

			T(3, 2) = 26 because there are 26 subpermutations on {1,2,3}, each of whose orbit is of size at most 2, namely:
Empty map, 1-->1, 1-->2, 1-->3, 2-->1, 2-->2, 2-->3, 3-->1, 3-->2, 3-->3, (1,2) --> (1,2), (1,3) --> (1,3), (2,3) --> (2,3), (1,2) --> (2,1), (1,3) --> (3,1), (2,3) --> (3,2), (1,2) --> (1,3), (1,3) --> (1,2), (2,3) --> (2,1), (1,2) --> (3,2), (1,3) --> (2,3), (2,3) --> (1,3), (1,2,3) --> (1,3,2), (1,2,3) --> (3,2,1), (1,2,3) --> (2,1,3), (1,2,3) --> (1,2,3).
Triangle starts:
1;
1, 2;
1, 4, 7;
1, 8, 26, 34;
1, 16, 115, 179, 209;
...

References

  • A. Laradji and A. Umar, On the number of subpermutations with fixed orbit size, Ars Combinatoria, 109 (2013), 447-460.

Crossrefs

Cf. A157400, A261762, A261764, A261765, A261766, A261767.

Formula

T(n,n) = A002720(n).
T(n,k) = Sum_{i=0..n} binomial(n,i)*A261762(n-i,k).
E.g.f. of column k: exp(Sum_{j=1..k} (j+1)*x^j/j).

Extensions

More terms from Alois P. Heinz, Oct 07 2015

A261765 Triangle read by rows: T(n,k) is the number of subpermutations of an n-set, whose orbits are each of size at most k with at least one orbit of size exactly k, and without fixed points. Equivalently, T(n,k) is the number of partial derangements of an n-set each of whose orbits is of size at most k with at least one orbit of size exactly k, and without fixed points.

Original entry on oeis.org

1, 1, 0, 1, 0, 3, 1, 0, 9, 8, 1, 0, 45, 32, 30, 1, 0, 165, 320, 150, 144, 1, 0, 855, 2240, 1800, 864, 840, 1, 0, 3843, 17360, 18900, 12096, 5880, 5760, 1, 0, 21819, 146048, 195300, 145152, 94080, 46080, 45360, 1, 0, 114075, 1256192, 2120580, 1959552, 1270080, 829440, 408240, 403200
Offset: 0

Views

Author

Samira Stitou, Sep 21 2015

Keywords

Comments

T(n,n) is A261766. Sum of rows is A144085.

Examples

			T(n,1) = 0 because there is no (partial) derangement with an orbit of size 1.
T(3,2) = 9 because there are 9 subpermutations on {1,2,3}, whose orbits are each of size at most 2 with at least one orbit of size exactly 2, and without fixed points, namely: (1 2 --> 2 1), (1 3 --> 3 1), (2 3 --> 3 2), (1-->2), (1-->3), (2-->1), (2-->3), (3-->1), (3-->2).
Triangle starts:
1;
1, 0;
1, 0, 3;
1, 0, 9, 8;
1, 0, 45, 32, 30;
1, 0, 165, 320, 150, 144;
1, 0, 855, 2240, 1800, 864, 840;
...

References

  • A. Laradji and A. Umar, On the number of subpermutations with fixed orbit size, Ars Combinatoria, 109 (2013), 447-460.

Crossrefs

Cf. A157400, A261762, A261763, A261764, A261766, A261767.

Formula

T(n,k) = A261762(n,k) - A261762(n,k-1).

Extensions

More terms from Alois P. Heinz, Nov 04 2015

A261766 a(n) is the number of partial derangements of an n-set with at least one orbit of size exactly n.

Original entry on oeis.org

1, 0, 3, 8, 30, 144, 840, 5760, 45360, 403200, 3991680, 43545600, 518918400, 6706022400, 93405312000, 1394852659200, 22230464256000, 376610217984000, 6758061133824000, 128047474114560000, 2554547108585472000, 53523844179886080000, 1175091669949317120000
Offset: 0

Views

Author

Samira Stitou, Sep 21 2015

Keywords

Examples

			a(3) = 8 because there are 8 partial derangements on {1,2,3} with at least one orbit of size 3 namely: (1,2) --> (2,3), (1,2)  --> (3,1), (1,3)  --> (2,1), (1,3) --> (3,2), (2,3)  --> (3,1), (2,3)  --> (1,2), (1,2,3) --> (2,3,1), (1,2,3)  --> (3,1,2).

References

  • A. Laradji and A. Umar, On the number of subpermutations with fixed orbit size, Ars Combinatoria, 109 (2013), 447-460.

Crossrefs

Cf. A001048, A059171, A157400, A261762, A261763, A261764, A261765, A261767.

Formula

a(n) = A261765(n,n) - A261765(n,n-1) for n>0, a(0)=1.

Extensions

More terms from Alois P. Heinz, Nov 04 2015
Showing 1-5 of 5 results.

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