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.

A261762 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, 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.

Original entry on oeis.org

1, 1, 1, 1, 1, 4, 1, 1, 10, 18, 1, 1, 46, 78, 108, 1, 1, 166, 486, 636, 780, 1, 1, 856, 3096, 4896, 5760, 6600, 1, 1, 3844, 21204, 40104, 52200, 58080, 63840, 1, 1, 21820, 167868, 363168, 508320, 602400, 648480, 693840, 1, 1, 114076, 1370268, 3490848, 5450400, 6720480
Offset: 0

Views

Author

Samira Stitou, Sep 21 2015

Keywords

Comments

The OEIS values correct the values b(n,k) in the Laradji-Umar Table 2.1 in column k=2. Note that the row sums (meaning: sums up to the diagonal of the triangle) in Table 2.1 in the article are also incorrect.
There were typos in the column (k=2) of the original article. The entry 94 should be 166 and the entry 784 should be 856, which have been corrected. Unlike most triangles the off-diagonal terms are not 0 because T(n, n)= T(n, n+k) for all nonnegative k which is obvious from the definition.

Examples

			T(3,2) = 10 because there are 10 subpermutations on {1,2,3}, each of whose orbit is of size at most 2, and without fixed points, namely: Empty map, (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, 1;
1, 1, 4;
1, 1, 10, 18;
1, 1, 46, 78, 108;
1, 1, 166, 486, 636, 780;
...

Links

Crossrefs

Cf. A157400, A261763, A261764, A261765, A261766, A261767.

Programs

  • Maple
    A261762 := proc(n,k)
    if k = 0 then
        1;
    else
        if k < 1 then
            g := 1;
        elif k < 2 then
            g := exp(x) ;
        else
            g := exp(x+add((j+1)*x^j/j,j=2..k)) ;
        fi;
        coeftayl(g,x=0,n) *n! ;
    end if;
end proc:
seq(seq( A261762(n,k),k=0..n),n=0..12) ; # R. J. Mathar, Nov 04 2015
  • Mathematica
    T[n_, k_] := SeriesCoefficient[ Exp[ x + Sum[ (j+1)*x^j/j, {j, 2, k}]], {x, 0, n}] * n!; Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-François Alcover, Apr 13 2017 *)

Formula

T(n,k) = T(n-1,k) + 3(n-1)T(n-2,k) + ... +(k+1)(n-1)(n-2)...(n-k+1)T(n-k,k) if k<=n.
T(n,k) = T(n,n) if k>n, not part of the triangle.
T(n,0) = T(n,1) = 1.
T(n,n) = A144085(n). (Diagonal)
G.f.: exp(x+(3x^2)/2+ ... +((k+1)x^k)/k).

Extensions

More terms from Alois P. Heinz, Oct 07 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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