login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A335212 a(n) = C(n) * d(n), where C(n) is the Catalan number and d(n) is the sorting probability of the Catalan poset P_n. 2
0, 1, 4, 8, 8, 9, 110, 572, 2496, 5762, 3254, 5020, 117912, 307819, 420394, 677350, 9611700, 58689330, 32290388, 430183260, 180484980, 5983159650, 44850171444, 120220997328, 235364198128, 135740049556, 2107819739960, 5252440129232, 37484769529504, 125244830989069 (list; graph; refs; listen; history; text; internal format)
OFFSET

2,3

COMMENTS

C(n) = A000108(n) is the Catalan number binomial(2n, n)/(n+1). P_n is the Catalan poset, which is the poset product of a 2-chain and an n-chain. The number of linear extensions of P_n is the Catalan number C(n). d(n) is Min_{x,y in P_n} |p_n(x<y) - p_n(x>y)|, where p_n(x<y) is the probability that x is less than y in the uniform random linear extension of P_n. It has been proved that a(n)/C(n) = O(n^{-5/4}), see the references: Chan, Pak, and Panova. In particular, we have a(n) < C(n).

LINKS

Swee Hong Chan, Table of n, a(n) for n = 2..1000

S. H. Chan, I. Pak, and G. Panova, Sorting probability of Catalan posets, preprint arXiv:2005.13686 [math.CO] (2020), 10 pp.

PROG

(Sage)

def a(n):

    lsp=[];

    minloc=[];

    for y in range (1, n+1):

        sum=0;

        for z in range (0, y+1):

            K=(binomial(2*y-z-1, y-1)*binomial(2*n-2*y+z, n-y+z)*(z)*(z+1)*(n+1))/(binomial(2*n, n)*(y)*(n-y+z+1));

            sum=sum+K;

            if 2*sum >= 1:

                h=sum-K;

                a_1=1-2*h;

                a_2=2*sum-1;

                if a_1< a_2:

                    minloc.append(x-1);

                    lsp.append((a_1)*(binomial(2*n, n)/(n+1)));

                else:

                    minloc.append(x);

                    lsp.append((a_2)*(binomial(2*n, n))/(n+1));

                break;

    return min(lsp)

CROSSREFS

Cf. A000108.

Sequence in context: A304651 A187768 A344984 * A141284 A272812 A273207

Adjacent sequences:  A335209 A335210 A335211 * A335213 A335214 A335215

KEYWORD

nonn

AUTHOR

Swee Hong Chan, May 26 2020

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified June 21 17:44 EDT 2021. Contains 345365 sequences. (Running on oeis4.)