A059946 -id:A059946

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A059443

Triangle T(n,k) (n >= 2, k = 3..n+floor(n/2)) giving number of bicoverings of an n-set with k blocks.

+10
27

1, 4, 4, 13, 39, 25, 3, 40, 280, 472, 256, 40, 121, 1815, 6185, 7255, 3306, 535, 15, 364, 11284, 70700, 149660, 131876, 51640, 8456, 420, 1093, 68859, 759045, 2681063, 3961356, 2771685, 954213, 154637, 9730, 105, 3280, 416560, 7894992, 44659776, 103290096

(list; graph; refs; listen; history; text; internal format)

OFFSET

2,2

REFERENCES

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 303, #40.

I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, John Wiley and Sons, N.Y., 1983.

LINKS

Alois P. Heinz, Rows n = 2..60, flattened

L. Comtet, Birecouvrements et birevêtements d’un ensemble fini, Studia Sci. Math. Hungar 3 (1968): 137-152. [Annotated scanned copy. Warning: the table of v(n,k) has errors.]

FORMULA

E.g.f. for m-block bicoverings of an n-set is exp(-x-1/2*x^2*(exp(y)-1))*Sum_{i=0..inf} x^i/i!*exp(binomial(i, 2)*y).

T(n, k) = Sum{j=0..n} Stirling2(n, j) * A060052(j, k). - David Pasino, Sep 22 2016

EXAMPLE

T(2,3) = 1: 1|12|2.

T(3,3) = 4: 1|123|23, 12|13|23, 12|123|3, 123|13|2.

T(3,4) = 4: 1|12|23|3, 1|13|2|23, 1|123|2|3, 12|13|2|3.

Triangle T(n,k) begins:

: 1;

: 4, 4;

: 13, 39, 25, 3;

: 40, 280, 472, 256, 40;

: 121, 1815, 6185, 7255, 3306, 535, 15;

: 364, 11284, 70700, 149660, 131876, 51640, 8456, 420;

: 1093, 68859, 759045, 2681063, 3961356, 2771685, 954213, 154637, 9730, 105;

...

MATHEMATICA

nmax = 8; imax = 2*(nmax - 2); egf := E^(-x - 1/2*x^2*(E^y - 1))*Sum[(x^i/i!)*E^(Binomial[i, 2]*y), {i, 0, imax}]; fx = CoefficientList[ Series[ egf , {y, 0, imax}], y]*Range[0, imax]!; row[n_] := Drop[ CoefficientList[ Series[fx[[n + 1]], {x, 0, imax}], x], 3]; Table[ row[n], {n, 2, nmax}] // Flatten (* Jean-François Alcover, Sep 21 2012 *)

PROG

(PARI) \ps 22;

s = 8; pv = vector(s); for(n=1, s, pv[n]=round(polcoeff(f(x, y), n, y)*n!));

for(n=1, s, for(m=3, poldegree(pv[n], x), print1(polcoeff(pv[n], m), ", "))) \\ Gerald McGarvey, Dec 03 2009

CROSSREFS

Columns k=3-10 give: A003462, A059945, A059946, A059947, A059948, A059949, A059950, A059951.

Row sums are A002718.

Main diagonal gives A275517.

Right border gives A275521.

KEYWORD

tabf,nonn,nice

AUTHOR

N. J. A. Sloane, Feb 01 2001

EXTENSIONS

More terms and additional comments from Vladeta Jovovic, Feb 14 2001

a(37) corrected by Gerald McGarvey, Dec 03 2009

STATUS

approved

A060093

Number of 5-block ordered bicoverings of an unlabeled n-set.

+10
4

0, 0, 0, 0, 125, 722, 2565, 7180, 17335, 37750, 76093, 144340, 260590, 451440, 755040, 1224964, 1935050, 2985380, 4509590, 6683720, 9736835, 13963670, 19739575, 27538060, 37951265, 51713706, 69729675, 93104700, 123181500

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,5

LINKS

Harry J. Smith, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

FORMULA

a(n) = binomial(n+9, 9) - 5*binomial(n+5, 5) - 10*binomial(n+3, 3) + 20*binomial(n+2, 2) + 30*binomial(n+1, 1) - 60*binomial(n, 0) + 24*binomial(n-1, -1).

G.f.: y^4*(-528*y + 125 + 970*y^2 - 980*y^3 + 570*y^4 - 180*y^5 + 24*y^6)/(-1 + y)^10.

E.g.f. for k-block ordered bicoverings of an unlabeled n-set is exp(-x - x^2/2*y/(1 - y))*Sum_{k>=0} 1/(1 - y)^binomial(k, 2)*x^k/k!.

a(n) = (n-1) *(n-2) *(n-3) *(n^6 + 51*n^5 + 1165*n^4 + 15885*n^3 + 130954*n^2 + 660504*n + 1451520)/ 362880, n > 0. - R. J. Mathar, Aug 10 2017

PROG

(PARI) a(n) = if(n<1, 0, binomial(n + 9, 9) - 5*binomial(n + 5, 5) - 10*binomial(n + 3, 3) + 20*binomial(n + 2, 2) + 30*binomial(n + 1, 1) - 60*binomial(n, 0) + 24*binomial(n - 1, -1)) \\ Harry J. Smith, Jul 01 2009

CROSSREFS

Column k=5 of A060092.

Cf. A059946, A060090, A060489.

KEYWORD

nonn,easy

AUTHOR

Vladeta Jovovic, Feb 26 2001

STATUS

approved

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