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A143672 Number of chains (totally ordered subsets) in the poset of Dyck paths ordered by inclusion.

%I A143672 #31 Jan 10 2015 08:59:11
%S A143672 1,2,4,24,816,239968,808814912,38764383658368,31491961129357837056,
%T A143672 503091371552266970507912704,179763631086276515267399940231898112,
%U A143672 1609791936564515363272979180683040232936253440
%N A143672 Number of chains (totally ordered subsets) in the poset of Dyck paths ordered by inclusion.
%C A143672 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:
%C A143672 n=0 1
%C A143672 n=1 1 + t
%C A143672 n=2 1 + 2t + t^2
%C A143672 n=3 1 + 5t + 9t^2 + 7t^3 + 2t^4
%C A143672 n=4 1 + 14t + 70t^2 + 176 t^3+249t^4 + 202t^5 + 88 t^6 + 16 t^7
%C A143672 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)
%C A143672 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).
%D A143672 R. P. Stanley, Enumerative Combinatorics 1, Cambridge University Press, New York, 1997.
%H A143672 J. Woodcock, <a href="http://garsia.math.yorku.ca/~zabrocki/dyckpathposet.html">Properties of the poset of Dyck paths ordered by inclusion</a>
%F A143672 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.
%e A143672 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.
%p A143672 d:= proc(x, y, l) option remember;
%p A143672       `if`(x=1, [[l[], y]], [seq(d(x-1, i, [l[], y])[], i=x-1..y)])
%p A143672     end:
%p A143672 le:= proc(l1, l2) local i;
%p A143672        for i to nops(l1) do if l1[i]>l2[i] then return false fi od;
%p A143672        true
%p A143672      end:
%p A143672 a:= proc(n) option remember; local l, m, g;
%p A143672       if n=0 then return 1 fi;
%p A143672       l:= d(n, n, []); m:= nops(l);
%p A143672       g:= proc(t) option remember;
%p A143672             1 +add(`if`(le(l[d], l[t]), g(d), 0), d=1..t-1);
%p A143672           end;
%p A143672       1+ add(g(h), h=1..m)
%p A143672     end:
%p A143672 seq(a(n), n=0..7);  # _Alois P. Heinz_, Jul 26 2011
%Y A143672 Cf. A143673, A143674, A193629. Chains on one vertex: A000108. Number of maximal chains: A005118.
%K A143672 nonn
%O A143672 0,2
%A A143672 Jennifer Woodcock (jennifer.woodcock(AT)ugdsb.on.ca), Aug 28 2008
%E A143672 a(6)-a(11), from _Alois P. Heinz_, Jul 26 2011, Aug 01 2011