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 A079267 d(n,s) = number of perfect matchings on {1, 2, ..., n} with s short pairs. 15
 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 (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 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 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). 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. 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. D. 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. 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): # 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 *) PROG (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 CROSSREFS Columns are A278990, A000806, A006198, A006199, A006200. Row sums are A001147. d(2n,n) gives A365744. Sequence in context: A342015 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 August 11 21:40 EDT 2024. Contains 375073 sequences. (Running on oeis4.)