A059074 Number of derangements of a multiset comprising 4 repeats of an n-element set.

1, 0, 1, 346, 748521, 3993445276, 45131501617225, 964363228180815366, 35780355973270898382001, 2158610844939711892526650456, 201028342764877992289387752167601, 27708893753238763155350683269145066450, 5459844285803153226360263675364357481841881

Offset: 0

Author

Barbara Haas Margolius (margolius(AT)math.csuohio.edu)

Keywords

• nonn
• nice

Comments

Previous name was: Card-matching numbers (Dinner-Diner matching numbers).

A deck has n kinds of cards, 4 of each kind. The deck is shuffled and dealt in to n hands with 4 cards each. A match occurs for every card in the j-th hand of kind j. A(n) is the number of ways of achieving no matches. The probability of no matches is A(n)/((4n)!/4!^n).

Number of fixed-point-free permutations of n distinct letters (ABCD...), each of which appears 4 times: 1111, 11112222, 111122223333, 1111222233334444, etc. If there is only one letter of each type we get A000166 - Zerinvary Lajos, Nov 05 2006

a(n) is the maximal number of totally mixed Nash equilibria in games of n players, each with 5 pure options. [Raimundas Vidunas, Jan 22 2014]

Examples

			There are 346 ways of achieving zero matches when there are 4 cards of each kind and 3 kinds of card so A(3)=346.
		

References

• F. N. David and D. E. Barton, Combinatorial Chance, Hafner, NY, 1962, Ch. 7 and Ch. 12.
• R.D. McKelvey and A. McLennan, The maximal number of regular totally mixed Nash equilibria, J. Economic Theory, 72:411-425, 1997.
• S. G. Penrice, Derangements, permanents and Christmas presents, The American Mathematical Monthly 98(1991), 617-620.
• J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, pp. 174-178.
• R. P. Stanley, Enumerative Combinatorics Volume I, Cambridge University Press, 1997, p. 71.

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..100
• Shalosh B. Ekhad, Christoph Koutschan, and Doron Zeilberger, There are EXACTLY 1493804444499093354916284290188948031229880469556 Ways to Derange a Standard Deck of Cards (ignoring suits) [and many other such useful facts], arXiv:2101.10147 [math.CO], 2021.
• Shalosh B. Ekhad, Terms and recurrences for multiset derangements.
• S. Even and J. Gillis, Derangements and Laguerre polynomials, Mathematical Proceedings of the Cambridge Philosophical Society, Volume 79, Issue 1, January 1976, pp. 135-143.
• F. F. Knudsen and I. Skau, On the Asymptotic Solution of a Card-Matching Problem, Mathematics Magazine 69 (1996), 190-197.
• Barbara H. Margolius, The Dinner-Diner Matching Problem, Mathematics Magazine, 76 (2003), 107-118.
• Barbara H. Margolius, Dinner-Diner Matching Probabilities
• Raimundas Vidunas, MacMahon's master theorem and totally mixed Nash equilibria, arXiv preprint arXiv:1401.5400 [math.CO], 2014.
• Raimundas Vidunas, Counting derangements and Nash equilibria Ann. Comb. 21, No. 1, 131-152 (2017).
• Index entries for sequences related to card matching

Crossrefs

Cf. A000166, A008290, A059056-A059071, A371252.

Programs

Maple

p := (x,k)->k!^2*sum(x^j/((k-j)!^2*j!),j=0..k); R := (x,n,k)->p(x,k)^n; f := (t,n,k)->sum(coeff(R(x,n,k),x,j)*(t-1)^j*(n*k-j)!,j=0..n*k); seq(f(0,n,4)/4!^n,n=0..18);

Mathematica

p[x_, k_] := k!^2*Sum[x^j/((k - j)!^2*j!), {j, 0, k}]; r[x_, n_, k_] := p[x, k]^n; f[t_, n_, k_] := Sum[Coefficient[r[x, n, k], x, j]*(t - 1)^j*(n*k - j)!, {j, 0, n*k}]; Table[f[0, n, 4]/4!^n, {n, 0, 18}] // Flatten (* Jean-François Alcover, Oct 21 2013, after Maple *)
Table[Integrate[Exp[-x] LaguerreL[4, x]^n, {x, 0, Infinity}], {n, 0, 16}] (* Jeremy Tan, Apr 25 2024 *)
rec = 3*(128*n^3 - 560*n^2 + 840*n - 537)*a[n] - n*(4096*n^6 - 24064*n^5 + 62720*n^4 - 92992*n^3 + 75248*n^2 - 38670*n + 4179)*a[n-1] - 2*n*(n-1)*(18432*n^5 - 99072*n^4 + 197120*n^3 - 191776*n^2 + 144568*n - 92531)*a[n-2] + 48*n*(n-1)*(n-2)*(768*n^4 - 2976*n^3 + 3104*n^2 - 2438*n + 1583)*a[n-3] + 288*n*(n-1)*(n-2)*(n-3)*(128*n^3 - 176*n^2 + 104*n - 129)*a[n-4] == 8192*n^6 - 28672*n^5 + 23680*n^4 - 7904*n^3 + 1416*n^2 + 14382*n - 1611;
RecurrenceTable[{rec, a[0] == 1, a[1] == 0, a[2] == 1, a[3] == 346}, a, {n, 0, 16}] (* Jeremy Tan, Apr 25 2024 *)

Python

def A059074(n):
    l = [1, 0, 1, 346]
    for k in range(4, n+1):
        num = (((((8192*k-28672)*k+23680)*k-7904)*k+1416)*k+14382)*k-1611 \
            + k*((((((4096*k-24064)*k+62720)*k-92992)*k+75248)*k-38670)*k+4179)*l[-1] \
            + 2*k*(k-1)*(((((18432*k-99072)*k+197120)*k-191776)*k+144568)*k-92531)*l[-2] \
            - 48*k*(k-1)*(k-2)*((((768*k-2976)*k+3104)*k-2438)*k+1583)*l[-3] \
            - 288*k*(k-1)*(k-2)*(k-3)*(((128*k-176)*k+104)*k-129)*l[-4]
        r = num // (3*(((128*k-560)*k+840)*k-537))
        l.append(r)
    return l[n] # Jeremy Tan, Apr 25 2024

Formula

G.f.: sum(coeff(R(x, n, k), x, j)*(t-1)^j*(n*k-j)!, j=0..n*k) where n is the number of kinds of cards, k is the number of cards of each kind (3 in this case) and R(x, n, k) is the rook polynomial given by R(x, n, k)=(k!^2*sum(x^j/((k-j)!^2*j!))^n (see Stanley or Riordan). coeff(R(x, n, k), x, j) indicates the coefficient for x^j of the rook polynomial.

From Jeremy Tan, Apr 25 2024: (Start)

a(n) = Integral_{x=0..oo} exp(-x)*L_4(x)^n dx, where L_n(x) is the Laguerre polynomial of degree n (Even and Gillis).

D-finite with recurrence 3*(128*n^3 - 560*n^2 + 840*n - 537)*a(n) - n*(4096*n^6 - 24064*n^5 + 62720*n^4 - 92992*n^3 + 75248*n^2 - 38670*n + 4179)*a(n-1) - 2*n*(n-1)*(18432*n^5 - 99072*n^4 + 197120*n^3 - 191776*n^2 + 144568*n - 92531)*a(n-2) + 48*n*(n-1)*(n-2)*(768*n^4 - 2976*n^3 + 3104*n^2 - 2438*n + 1583)*a(n-3) + 288*n*(n-1)*(n-2)*(n-3)*(128*n^3 - 176*n^2 + 104*n - 129)*a(n-4) = 8192*n^6 - 28672*n^5 + 23680*n^4 - 7904*n^3 + 1416*n^2 + 14382*n - 1611 (Ekhad).

a(n) ~ A014608(n)/exp(4) ~ n^(4*n)*(32/3)^n*sqrt(8*Pi*n)/exp(4*n+4). (End)

Extensions

Name changed by Jeremy Tan, Apr 25 2024