Alexander Karpov - cp's OEIS frontend

A307416 Number of narcissistic group-separable preference profiles.

1, 1, 1, 6, 144, 13440, 4976640, 7390494720, 44033145569280, 1050540969012756480, 100246620373125798297600, 38247242364455762072803737600, 58346834889692144013481754807500800, 355946631376225773365104516210274756198400

Offset: 0

Alexander Karpov, Apr 07 2019

nonn

A. Karpov, On the Number of Group-Separable Preference Profiles, Group Decision and Negotiation, 2019, pp. 1-17.

a(n) = {j=1;if(n>1,j=sum(k=1,n-1,binomial(n-1, k-1)*2^(2*n*k-2*k^2-n)*a(k)*a(n-k)));j;} \\ Jinyuan Wang, Apr 08 2019

a(n) = Sum_{k=1..n-1} binomial(n-1, k-1)*2^(2*n*k-2*k^2-n)*a(k)*a(n-k).

More terms from Jinyuan Wang, Apr 08 2019

A294085 a(n) is the number of self-symmetric anonymous and neutral equivalence classes of preference profiles with 3 alternatives and n agents (IANC model).

Original entry on oeis.org

1, 1, 3, 4, 8, 10, 17, 20, 30, 35, 49, 56, 75, 84, 108, 120, 150, 165, 202, 220, 264, 286, 338, 364, 425, 455, 525, 560, 640, 680, 771, 816, 918, 969, 1083, 1140, 1267, 1330, 1470, 1540, 1694, 1771, 1940, 2024, 2208, 2300, 2500, 2600, 2817, 2925, 3159, 3276, 3528, 3654, 3925

Offset: 0

Alexander Karpov, Apr 12 2018

Colin Barker, Table of n, a(n) for n = 0..1000
A. Karpov, An Informational Basis for Voting Rules, NRU Higher School of Economics. Series WP BRP "Economics/EC". 2018. No. 188
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1,1,-1,-2,2,1,-1).

Cf. A037240, A005513.

For odd n, it is A000292.

Vec((1 + x^3 + x^4) / ((1 - x)^4*(1 + x)^3*(1 - x + x^2)*(1 + x + x^2)) + O(x^60)) \\ Colin Barker, May 11 2018

a(n) = 2*A005513(n-6) - A037240(n).
If n is odd, a(n) = (n+5)*(n+3)*(n+1)/48;
If n is even, a(n) = ceiling((n+4)^2*(n+2)/48).
From Colin Barker, May 11 2018: (Start)
G.f.: (1 + x^3 + x^4) / ((1 - x)^4*(1 + x)^3*(1 - x + x^2)*(1 + x + x^2)).
a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) + a(n-6) - a(n-7) - 2*a(n-8) + 2*a(n-9) + a(n-10) - a(n-11) for n>10.
(End)

A296506 Number of generalized knockout tournament seedings with 3 players in each match and n rounds.

Original entry on oeis.org

1, 280, 833712928048000000

Offset: 1

Alexander Karpov, Dec 13 2017

nonn

Next term is too large to include.

Alexander Karpov, Generalized knockout tournaments, National Research University Higher School of Economics. WP7/2017/03.
Alexander Karpov, Generalized knockout tournaments.

Cf. A067667 (number of knockout tournament seedings with two players in each match).

f[n_] := (3^n)!/6^((3^n - 1)/2); Array[f, 3] (* Robert G. Wilson v, Dec 27 2017 *)

a(n) = (3^n)!/6^((3^n-1)/2).

A296260 Number of preference profiles with 4 alternatives and n agents (IANC model).

Original entry on oeis.org

1, 1, 17, 111, 762, 4095, 19941, 84825, 329214, 1168740, 3858348, 11920740, 34773590, 96282900, 254473884, 644637204, 1571330916, 3697182450, 8421423582, 18615637950, 40023753924, 83859017814, 171530071362, 343059613650, 671825586021, 1289904147324, 2430974136780

Offset: 1

Alexander Karpov, Dec 15 2017

nonn

Ö. Egecioglu, Uniform generation of anonymous and neutral preference profiles for social choice rules, Monte Carlo Methods and Applications, 15(3), Jan 2009, 241-255.

Cf. A037240 for 3 alternatives.

Array[Binomial[# + 23, 23]/24 + Which[Divisible[#1, 12], 3 Binomial[#1/2 + 11, 11]/8 + Binomial[#1/3 + 7, 7]/3 + Binomial[#1/4 + 5, 5]/4, MemberQ[{1, 5, 7, 11}, #2], 0, MemberQ[{2, 10}, #2], 3 Binomial[#1/2 + 11, 11]/8, MemberQ[{3, 9}, #2], Binomial[#1/3 + 7, 7]/3, MemberQ[{4, 8}, #2], 3 Binomial[#1/2 + 11, 11]/8 + Binomial[#1/4 + 5, 5]/4, True, 3 Binomial[#1/2 + 11, 11]/8 + Binomial[#1/3 + 7, 7]/3 ] & @@ {#, Mod[#, 12]} &, 26] (* Michael De Vlieger, Dec 18 2017 *)

if n ==  0  mod 12, a(n) = C(n+23,23)/24 + C(n/2+11,11)*3/8 + C(n/3+7,7)/3+C(n/4+5,5)/4;
if n ==  1,5,7,11  mod 12, a(n) = C(n+23,23)/24;
if n == 2,10 mod 12, a(n) = C(n+23,23)/24 + C(n/2+11,11)*3/8;
if n == 3,9 mod 12, a(n) = C(n+23,23)/24  + C(n/3+7,7)/3;
if n == 4,8 mod 12, a(n) = C(n+23,23)/24 + C(n/2+11,11)*3/8 +C(n/4+5,5)/4;
if n == 6 mod 12, a(n) = C(n+23,23)/24 + C(n/2+11,11)*3/8 + C(n/3+7,7)/3.

More terms from Michael De Vlieger, Dec 18 2017

A289747 Erroneous version of A036981.

Original entry on oeis.org

1, 6, 720, 120960, 2264371200, 48920701648896000, 1570969121006520000000000000

Offset: 1

Alexander Karpov, Jul 11 2017

dead

Previous incorrect name was "Number of round-robin tournament schedules with 2*n participants."

A261187 a(n) = (2^(n-1))!y(n) where y(n)=1/2(y(n-1))^2+1 for n>=2 and y(1)=1.

Original entry on oeis.org

1, 3, 51, 131355, 131953155208875, 5496027066067360087228913484456796875, 27805296606704951937976342299927372748633425216234990144120838935506416477839670037841796875

Offset: 1

Alexander Karpov, Aug 11 2015

nonn

a(n) is also the number of knockout tournament seedings that satisfy the symmetry property.

Alexander Karpov, A theory of knockout tournament seedings, Heidelberg University, AWI Discussion Paper Series, No. 600.

Cf. A067667 (number of seedings).

Table[(2^(n-1))!*FoldList[(1/2)*(#1)^2+1&,1,Range[2,7]][[n]],{n,1,7}] (* Ivan N. Ianakiev, Aug 25 2015 *)

A261125 a(n) = (2^(n-1))!*a(n-1) for n>=1, a(0) = 1.

Original entry on oeis.org

1, 1, 2, 48, 1935360, 40493130637639680000, 10654991354747516157752604498631700563938508800000000000

Offset: 0

Alexander Karpov, Aug 09 2015

nonn

The next term is too large to display.
The number of knockout tournament seedings that satisfy the delayed confrontation property.
a(n) is the number of permutations p of [2^n] such that {p(1),...,p(2^k)} = [2^k] for all k = 0..n:  a(2) = 2: 1234, 1243. - Alois P. Heinz, Feb 04 2023

Michel Marcus, Table of n, a(n) for n = 0..9
Alexander Karpov, A theory of knockout tournament seedings, Heidelberg University, AWI Discussion Paper Series, No. 600.

Cf. A000722, A067667 (number of seedings).

[n le 1 select n else Self(n-1)*Factorial(2^(n - 1)): n in [1..7]]; // Vincenzo Librandi, Aug 10 2015

a:= proc(n) option remember:
      `if`(n=0, 1, a(n-1)*(2^(n-1))!)
    end:
seq(a(n), n=0..6);  # Alois P. Heinz, Feb 04 2023

RecurrenceTable[{a[1] == 1, a[n] == a[n-1] (2^(n - 1))!}, a, {n, 10}] (* Vincenzo Librandi, Aug 10 2015 *)
FoldList[(2^#2)!*#1&,1,Range@6] (* Ivan N. Ianakiev, Aug 10 2015 *)

first(m)=my(v=vector(m));v[1]=1;for(i=2,m,v[i]=(2^(i-1))!*v[i-1];);v; \\ Anders Hellström, Aug 10 2015

a(n) = Product_{j=0..n-1} (2^j)!. - Alois P. Heinz, Feb 04 2023

a(0)=1 prepended by Alois P. Heinz, Feb 04 2023

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User: Alexander Karpov

A307416 Number of narcissistic group-separable preference profiles.

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A294085 a(n) is the number of self-symmetric anonymous and neutral equivalence classes of preference profiles with 3 alternatives and n agents (IANC model).

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A296506 Number of generalized knockout tournament seedings with 3 players in each match and n rounds.

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A296260 Number of preference profiles with 4 alternatives and n agents (IANC model).

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A289747 Erroneous version of A036981.

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A261187 a(n) = (2^(n-1))!*y(n) where y(n)=1/2*(y(n-1))^2+1 for n>=2 and y(1)=1.

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A261125 a(n) = (2^(n-1))!*a(n-1) for n>=1, a(0) = 1.

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A261187 a(n) = (2^(n-1))!y(n) where y(n)=1/2(y(n-1))^2+1 for n>=2 and y(1)=1.