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 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 (list; graph; refs; listen; history; text; internal format)
 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 LINKS Alois P. Heinz, Rows n = 0..5, flattened Wikipedia, Chromatic Polynomial 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 Adjacent sequences: A141619 A141620 A141621 * A141623 A141624 A141625 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

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Last modified December 7 08:58 EST 2022. Contains 358653 sequences. (Running on oeis4.)