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A335213
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a(n) = C(n)*b(n), where C(n) is the n-th Catalan number and b(n) is the probability p_n(x<y) closest to 1/2 with x,y taken from the Catalan poset P_n.
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1
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1, 2, 5, 25, 70, 210, 660, 2145, 7150, 32274, 105633, 368940, 1278264, 5001332, 17889032, 64483720, 234013500, 854286930, 3265915016, 12018041880, 45831524310, 168538227000, 622526987940, 2491083699390, 9301358635140, 34834645482780, 130820066005200, 503747328390300
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OFFSET
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2,2
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COMMENTS
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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).
p_n(x,y) for x,y in P_n is the probability that x is less than y in the uniform random linear extension of P_n.
b(n) is the probability p_n(x<y) that is closest to 1/2 for all x,y in P_n.
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LINKS
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PROG
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(Sage)
def a(n):
lsp=[];
chalf=[];
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:
chalf.append(h);
lsp.append(a_1);
else:
chalf.append(sum);
lsp.append(a_2);
break;
yindex=lsp.index(min(lsp));
return (chalf[yindex])*(binomial(2*n, n))/(n+1)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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