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%I #16 Dec 28 2014 17:45:20
%S 1,1,1,-1,1,-5,10,-9,3,1,-21,210,-1321,5823,-18968,46908,-89034,
%T 129490,-142270,114532,-63791,21940,-3499,1,-84,3486,-95228,1924965,
%U -30690520,401700964,-4436161044,42161182074,-350011820616,2567538234448
%N Triangle read by rows: coefficients of chromatic polynomials for the poset of Dyck paths ordered by inclusion.
%C Number of entries in the rows are the Catalan numbers, see A000108.
%D G. Berman and K. D. Fryer, Introduction to Combinatorics, Academic Press, New York, 1972.
%D R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999
%H Alois P. Heinz, <a href="/A141622/b141622.txt">Rows n = 0..5, flattened</a>
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Chromatic_polynomial">Chromatic Polynomial</a>
%H J. Woodcock, <a href="http://garsia.math.yorku.ca/~zabrocki/papers/DPfinal.pdf">Properties of the poset of Dyck paths ordered by inclusion</a>
%e Chromatic polynomial for D_3 is t^5 - 5t^4 + 10t^3 - 9t^2 +3t =>
%e [1, -5, 10, -9, 3]
%e Triangle begins:
%e 1;
%e 1;
%e 1, -1;
%e 1, -5, 10, -9, 3;
%e 1, -21, 210, -1321, 5823, -18968, 46908, ...
%e 1, -84, 3486, -95228, 1924965, -30690520, 401700964, ...
%p with(networks);
%p new(G); # this is the graph for D_3
%p addvertex({1, 2, 3, 4}, G); addedge(Cycle(1, 2, 3, 4), G);
%p addvertex(5, G); addedge({4, 5}, G); draw(G);
%p ans:= sort (expand (chrompoly(G, x)));
%p # 2nd program
%p with(networks):
%p d:= proc(x, y, l) option remember;
%p `if`(x<=1, [[l[], y]], [seq(d(x-1, i, [l[], y])[], i=x-1..y)])
%p end:
%p le:= proc(l1, l2) local i;
%p for i to nops(l1) do if l1[i]>l2[i] then return false fi od;
%p true
%p end:
%p T:= proc(n) local l, m, p;
%p l:= d(n, n, []); m:= nops(l);
%p p:= chrompoly(graph({$1..m}, {seq(seq(`if`(le(l[i], l[j]),
%p `if`(true in {seq(k<>i and k<>j and le(l[i], l[k])
%p and le(l[k], l[j]), k=1..m)}, NULL, {i, j}), NULL),
%p j=i+1..m), i=1..m)}), t);
%p seq(coeff(p, t, m-i), i=0..m-1)
%p end:
%p seq(T(n), n=0..4); # _Alois P. Heinz_, Jul 24 2011
%Y Cf. A000108.
%K sign,tabf
%O 0,6
%A Jennifer Woodcock (Jennifer.Woodcock(AT)ugdsb.on.ca), Aug 23 2008
%E More terms from _Alois P. Heinz_, Jul 24 2011