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A159233
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Number of edge colorings of the Petersen graph.
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3
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0, 0, 0, 0, 49920, 18936480, 1293857280, 34839270480, 518880929280, 5122755187200, 37376074229760, 216261381584160, 1041571537839360, 4323020652281760, 15864787390932480, 52496402135289840, 159036030441200640, 446469968164496640, 1172916400659517440
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OFFSET
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0,5
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COMMENTS
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The Petersen graph G is a cubic symmetric graph on 10 vertices and 15 edges with edge chromatic number 4. a(n) is also the number of (vertex) n-colorings of L(G), the line graph (or interchange graph) of G.
"An edge coloring of a graph G is a coloring of the edges of G such that adjacent edges (or the edges bounding different regions) receive different colors. An edge coloring containing the smallest possible number of colors for a given graph is known as a minimum edge coloring." - Eric Weisstein, Edge Coloring.
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LINKS
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Timme, Marc; van Bussel, Frank; Fliegner, Denny; Stolzenberg, Sebastian (2009) "Counting complex disordered states by efficient pattern matching: chromatic polynomials and Potts partition functions", New J. Phys. 11 023001, doi: 10.1088/1367-2630/11/2/023001.
Index entries for linear recurrences with constant coefficients, signature (16, -120, 560, -1820, 4368, -8008, 11440, -12870, 11440, -8008, 4368, -1820, 560, -120, 16, -1).
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FORMULA
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a(n) = n^15 -30*n^14 + ... (see Maple program).
G.f.: 240 * x^4 * (120539*x^11 +4939568*x^10 +71258450*x^9 +441713760*x^8 +1285299570*x^7 +1834236432*x^6 +1296079344*x^5 +442507920*x^4 +68258235*x^3 +4153600*x^2 +75574*x +208) / (x-1)^16. - Colin Barker, Nov 28 2012
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MAPLE
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a:= n-> n^15 -30*n^14 +425*n^13 -3780*n^12 +23658*n^11 -110594*n^10 +399500*n^9 -1136005*n^8 +2560246*n^7 -4553907*n^6 +6285354*n^5 -6504300*n^4 +4739880*n^3 -2156064*n^2 +455616*n: seq(a(n), n=0..20);
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MATHEMATICA
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Table[n^15 - 30 n^14 + 425 n^13 - 3780 n^12 + 23658 n^11 - 110594 n^10 + 399500 n^9 - 1136005 n^8 + 2560246 n^7 - 4553907 n^6 + 6285354 n^5 - 6504300 n^4 + 4739880 n^3 - 2156064 n^2 + 455616 n, {n, 0, 18}] (* Michael De Vlieger, Mar 27 2016 *)
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PROG
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(PARI) a(n)=prod(i=0, 3, n-i)*(n^11 -24*n^10 +270*n^9 -1890*n^8 +9204*n^7 -32960*n^6 +89156*n^5 -183285*n^4 +282060*n^3 -310476*n^2 +220128*n -75936) \\ Charles R Greathouse IV, Nov 28 2012
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CROSSREFS
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Cf. A270842 for edge colorings under the automorphisms of the Petersen graph. A296913.
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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