A079267 d(n,s) = number of perfect matchings on {1, 2, ..., n} with s short pairs.

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, 70371, 8106, 666, 36, 1

Offset: 0

Author

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

Keywords

• easy
• nice
• nonn
• tabl

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

This is also the number of configurations of n indistinguishable pairs placed on the vertices of the ladder graph P_1 X P_2n (i.e., a path of length 2n) such that s such pairs are joined by an edge; equivalently the number of "s-domino" configurations in the game of memory played on a 1 X 2n rectangular array, see [Young]. - Donovan Young, Oct 23 2018

Examples

			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)).
		

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

• Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows 0 <= n <= 150, flattened).
• Marilena Barnabei, Niccolò Castronuovo, and Matteo Silimbani, Hertzsprung patterns on involutions, arXiv:2412.03449 [math.CO], 2024. See p. 10.
• Naiomi T. Cameron and Kendra Killpatrick, Statistics on Linear Chord Diagrams, arXiv:1902.09021 [math.CO], 2019.
• E. S. Krasko, I.N. Labutin, and A. V. Omelchenko, Enumeration of labeled and unlabeled Hamiltonian cycles in complete k-partite graphs, J. Math. Sci. 255 (2021) 71-87, eq (5).
• 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)
• Avichai Marmor, Schur-Positivity of Short Chords in Matchings, arXiv:2307.09894 [math.CO], 2023.
• Jeremy L. Martin, The slopes determined by n points in the plane, arXiv:math/0302106 [math.AG], 2003-2006.
• Jeremy L. Martin, The slopes determined by n points in the plane, Duke Math. J., Volume 131, Number 1 (2006), 119-165.
• Donovan Young, The Number of Domino Matchings in the Game of Memory, Journal of Integer Sequences, Vol. 21 (2018), Article 18.8.1.
• Donovan Young, Generating Functions for Domino Matchings in the 2 * k Game of Memory, arXiv:1905.13165 [math.CO], 2019. Also in J. Int. Seq., Vol. 22 (2019), Article 19.8.7.
• Donovan Young, Bubbles in Linear Chord Diagrams: Bridges and Crystallized Diagrams, arXiv:2408.17232 [math.CO], 2024. See p. 18.

Crossrefs

Columns are A278990, A000806, A006198, A006199, A006200.

Row sums are A001147.

d(2n,n) gives A365744.

Programs

Maple

d := (n,s) -> 1/s! * sum('((-1)^(h-s)*(2*n-h)!/(2^(n-h)*(n-h)!*(h-s)!))','h'=s..n):
# alternative by R. J. Mathar, Aug 19 2022
A079267 := proc(n,k)
    option remember ;
    if n =0 and k =0 then
        1;
    elif k > n or k < 0 then
        0;
    else
        procname(n-1,k-1)+(2*n-2-k)*procname(n-1,k)+(k+1)*procname(n-1,k+1) ;
    end if;
end proc:
seq(seq( A079267(n,k),k=0..n),n=0..13) ;

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 *)

PARI

{T(n, k) = 2^(k-n)*binomial(n,k)*hyperu(k-n, k-2*n, -2)};
for(n=0,10, for(k=0,n, print1(round(T(n,k)), ", "))) \\ G. C. Greubel, Apr 10 2019

Sage

[[2^(k-n)*binomial(n,k)*hypergeometric_U(k-n,k-2*n,-2).simplify_hypergeometric() for k in (0..n)] for n in (0..10)] # G. C. Greubel, Apr 10 2019

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

Extensions

Extra terms added by Paul Barry, Nov 25 2009