login
A141622
Triangle read by rows: coefficients of chromatic polynomials for the poset of Dyck paths ordered by inclusion.
1
1, 1, 1, -1, 1, -5, 10, -9, 3, 1, -21, 210, -1321, 5823, -18968, 46908, -89034, 129490, -142270, 114532, -63791, 21940, -3499, 1, -84, 3486, -95228, 1924965, -30690520, 401700964, -4436161044, 42161182074, -350011820616, 2567538234448
OFFSET
0,6
COMMENTS
Number of entries in the rows are the Catalan numbers, see A000108.
REFERENCES
G. Berman and K. D. Fryer, Introduction to Combinatorics, Academic Press, New York, 1972.
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999
EXAMPLE
Chromatic polynomial for D_3 is t^5 - 5t^4 + 10t^3 - 9t^2 +3t =>
[1, -5, 10, -9, 3]
Triangle begins:
1;
1;
1, -1;
1, -5, 10, -9, 3;
1, -21, 210, -1321, 5823, -18968, 46908, ...
1, -84, 3486, -95228, 1924965, -30690520, 401700964, ...
MAPLE
with(networks);
new(G); # this is the graph for D_3
addvertex({1, 2, 3, 4}, G); addedge(Cycle(1, 2, 3, 4), G);
addvertex(5, G); addedge({4, 5}, G); draw(G);
ans:= sort (expand (chrompoly(G, x)));
# 2nd program
with(networks):
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:
T:= proc(n) local l, m, p;
l:= d(n, n, []); m:= nops(l);
p:= chrompoly(graph({$1..m}, {seq(seq(`if`(le(l[i], l[j]),
`if`(true in {seq(k<>i and k<>j and le(l[i], l[k])
and le(l[k], l[j]), k=1..m)}, NULL, {i, j}), NULL),
j=i+1..m), i=1..m)}), t);
seq(coeff(p, t, m-i), i=0..m-1)
end:
seq(T(n), n=0..4); # Alois P. Heinz, Jul 24 2011
CROSSREFS
Cf. A000108.
Sequence in context: A280943 A316707 A109360 * A144136 A343852 A198286
KEYWORD
sign,tabf
AUTHOR
Jennifer Woodcock (Jennifer.Woodcock(AT)ugdsb.on.ca), Aug 23 2008
EXTENSIONS
More terms from Alois P. Heinz, Jul 24 2011
STATUS
approved