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A143672
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Number of chains (totally ordered subsets) in the poset of Dyck paths ordered by inclusion.
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4
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1, 2, 4, 24, 816, 239968, 808814912, 38764383658368, 31491961129357837056, 503091371552266970507912704, 179763631086276515267399940231898112, 1609791936564515363272979180683040232936253440
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OFFSET
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0,2
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COMMENTS
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For each n, each vertex in the Hasse diagram of the poset corresponds to a Dyck path of size n. The chain polynomial, C(P,t)=(1 + sum_k(c_k*t^(k+1)), gives the breakdown of this sequence by number of vertices in the chain. The 1 in front of the sum in this equation denotes the empty chain. The coefficient, c_k, gives the number of chains of length k and the exponent, (k+1), indicates the number of vertices in the chain. Here are the chain polynomials corresponding to this sequence:
n=0 1
n=1 1 + t
n=2 1 + 2t + t^2
n=3 1 + 5t + 9t^2 + 7t^3 + 2t^4
n=4 1 + 14t + 70t^2 + 176 t^3+249t^4 + 202t^5 + 88 t^6 + 16 t^7
n=5 1 + 42t + 552t^2 + 3573t^3 + 13609t^4+ 33260 t^5 + 54430t^6 + 60517t^7 + 45248t^8 + 21824t^9 + 6144t^(10) + 768t^(11)
Note that for each n, the coefficient of t is equal to the Catalan number, C_n. It is well-known that the number of Dyck paths in D_n is given by C_n (A000108). The coefficient in front of the largest power of t gives the number of maximal (and also maximum) chains (A005118).
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REFERENCES
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R. P. Stanley, Enumerative Combinatorics 1, Cambridge University Press, New York, 1997.
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LINKS
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FORMULA
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a(n) = 1 + Sum_{i,j in {1..C(n)}} (2*delta - zeta)^(-1)[i,j] where delta is the identity matrix and the zeta matrix is defined: zeta[a,b] = 1 if a<=b in D_n and 0 otherwise.
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EXAMPLE
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a(3) = 24 since in D_3 there are 2 chains of length 3 (i.e., on 4 vertices in the Hasse diagram), 7 chains of length 2 (on 3 vertices), 9 chains of length 1 (on 2 vertices), 5 chains of length 0 (on 1 vertex) and the empty chain: 2 + 7 + 9 + 5 + 1 = 24 chains.
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MAPLE
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d:= proc(x, y, l) option remember;
`if`(x=1, [[l[], y]], [seq(d(x-1, i, [l[], y])[], i=x-1..y)])
end:
le:= proc(l1, l2) local i;
for i to nops(l1) do if l1[i]>l2[i] then return false fi od;
true
end:
a:= proc(n) option remember; local l, m, g;
if n=0 then return 1 fi;
l:= d(n, n, []); m:= nops(l);
g:= proc(t) option remember;
1 +add(`if`(le(l[d], l[t]), g(d), 0), d=1..t-1);
end;
1+ add(g(h), h=1..m)
end:
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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Jennifer Woodcock (jennifer.woodcock(AT)ugdsb.on.ca), Aug 28 2008
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EXTENSIONS
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STATUS
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approved
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