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A079267 d(n,s) = number of perfect matchings on {1, 2, ..., n} with k short pairs. 3
1, 0, 1, 1, 1, 1, 5, 6, 3, 1, 36, 41, 21, 6, 1, 329, 365, 185, 55, 10, 1, 3655, 3984, 2010, 610, 120, 15, 1, 47844, 51499, 25914, 7980, 1645, 231, 21, 1, 721315, 769159, 386407, 120274, 25585, 3850, 406, 28, 1, 12310199, 13031514, 6539679, 2052309, 446544 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,7

COMMENTS

Read backwards, the n-th row of the triangle gives the Hilbert series of the variety of slopes determined by n points in the plane.

From Paul Barry, Nov 25 2009: (Start)

Reversal of coefficient array for the polynomials P(n,x)= Sum_{k=0..n} (C(n+k,2k)(2k)!/(2^k*k!))*x^k*(1-x)^(n-k).

Note that P(n,x) = Sum_{k=0..n} A001498(n,k)*x^k*(1-x)^(n-k). (End)

Equivalent to the original definition: Triangle of fixed-point free involutions on [1..2n] (=A001147) by number of cycles with adjacent integers. - Olivier Gérard, Mar 23 2011

Conjecture: Asymptotically, the n-th row has a Poisson distribution with mean 1. - David Callan, Nov 11 2012

REFERENCES

G. Kreweras and Y. Poupard, Sur les partitions en paires d'un ensemble fini totalement ordonné, Publications de l'Institut de Statistique de l'Université de Paris, 23 (1978), 57-74.

LINKS

Table of n, a(n) for n=0..49.

G. Kreweras and Y. Poupard, Sur les partitions en paires d'un ensemble fini totalement ordonné, Publications de l'Institut de Statistique de l'Université de Paris, 23 (1978), 57-74. (Annotated scanned copy)

J. L. Martin, The slopes determined by n points in the plane, arXiv:math/0302106 [math.AG], 2003-2006.

J. L. Martin, The slopes determined by n points in the plane, Duke Math. J., Volume 131, Number 1 (2006), 119-165.

FORMULA

d(n, s) = 1/s! * Sum_{h=s..n}(((-1)^(h-s)*(2*n-h)!/(2^(n-h)*(n-h)!*(h-s)!))).

E.g.f.: exp((x-1)*(1-sqrt(1-2*y)))/sqrt(1-2*y). - Vladeta Jovovic, Dec 15 2008

EXAMPLE

Triangle begins:

   1

   0  1

   1  1  1

   5  6  3 1

  36 41 21 6 1

From Paul Barry, Nov 25 2009: (Start)

Production matrix begins

       0,      1,

       1,      1,      1,

       4,      4,      2,     1,

      18,     18,      9,     3,     1,

      96,     96,     48,    16,     4,    1,

     600,    600,    300,   100,    25,    5,   1,

    4320,   4320,   2160,   720,   180,   36,   6,  1,

   35280,  35280,  17640,  5880,  1470,  294,  49,  7, 1,

  322560, 322560, 161280, 53760, 13440, 2688, 448, 64, 8, 1

Complete this by adding top row (1,0,0,0,....) and take inverse: we obtain

   1,

   0,  1,

  -1, -1,  1,

  -2, -2, -2,  1,

  -3, -3, -3, -3,  1,

  -4, -4, -4, -4, -4,  1,

  -5, -5, -5, -5, -5, -5,  1,

  -6, -6, -6, -6, -6, -6, -6,  1,

  -7, -7, -7, -7, -7, -7, -7, -7,  1,

  -8, -8, -8, -8, -8, -8, -8, -8, -8,  1 (End)

The 6 involutions with no fixed point on [1..6] with only one 2-cycle with adjacent integers are ((1, 2), (3, 5), (4, 6)), ((1, 3), (2, 4), (5, 6)), ((1, 3), (2, 6), (4, 5)), ((1, 5), (2, 3), (4, 6)), ((1, 5), (2, 6), (3, 4)), and ((1, 6), (2, 5), (3, 4)).

MAPLE

d := (n, s) -> 1/s! * sum('((-1)^(h-s)*(2*n-h)!/(2^(n-h)*(n-h)!*(h-s)!))', 'h'=s..n):

MATHEMATICA

nmax = 9; d[n_, s_] := (2^(s-n)*(2n-s)!* Hypergeometric1F1[s-n, s-2n, -2])/ (s!*(n-s)!); Flatten[ Table[d[n, s], {n, 0, nmax}, {s, 0, n}]] (* Jean-François Alcover, Oct 19 2011, after Maple *)

CROSSREFS

Row sums are A001147. Columns are A000806, A006198, A006199, A006200.

Sequence in context: A106599 A222466 A195448 * A060296 A114598 A272489

Adjacent sequences:  A079264 A079265 A079266 * A079268 A079269 A079270

KEYWORD

easy,nice,nonn,tabl

AUTHOR

Jeremy Martin (martin(AT)math.umn.edu), Feb 05 2003

EXTENSIONS

Extra terms added by Paul Barry, Nov 25 2009

STATUS

approved

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Last modified February 24 11:12 EST 2018. Contains 299603 sequences. (Running on oeis4.)