0,5

Fendley and Krushkal: "One of the remarkable features of the chromatic polynomial chi(Q) is Tutte's golden identity. This relates chi(phi+2) for any triangulation of the sphere to (chi(phi+1))^2 for the same graph, where phi denotes the golden ratio. We show that this result fits in the framework of quantum topology and give a proof of Tutte's identity using the notion of the chromatic algebra, whose Markov trace is the chromatic polynomial of an associated graph. We also show that another relation of Tutte's for the chromatic polynomial at Q=phi+1 precisely corresponds to a Jones-Wenzl projector in the Temperley-Lieb algebra. We show that such a relation exists whenever Q = 2+2cos(2 pi j/(n+1)) for j<n positive integers. When j=1, these are the Beraha numbers and in this case the existence of such a relation was conjectured by Tutte. We present a recursive formula for this sequence of chromatic polynomial relations. - Jonathan Vos Post (jvospost3(AT)gmail.com), Nov 04 2007

Beraha, S., Ph.D. thesis. Baltimore, MD: Johns Hopkins University, 1974.

S. R. Finch, Mathematical Constants (Encyclopedia of Mathematics and its Applications), chapter 5.25

Tutte, W. T., "More about Chromatic Polynomials and the Golden Ratio." In Combinatorial Structures and their Applications: Proc. Calgary Internat. Conf., Calgary, Alberta, 1969. New York: Gordon and Breach, p. 439, 1969.

Paul Fendley and Vyacheslav Krushkal, Tutte chromatic identities and the Temperley-Lieb algebra, Nov 1, 2007.

Eric Weisstein's World of Mathematics, Beraha Constants.

f(n,x)=(2x-1)f(n-1,x)-x^2*f(n-2,x), where f(n,x) is the monic characteristic polynomial of the n X n matrix from the definition and f(0,x)=1.

f(n,x)=(2x-1)f(n-1,x)-x^2*f(n-2,x), where f(n,x) is the monic characteristic polynomial of the n X n matrix from the definition and f(0,x)=1. See formula in Fendley and Krushkal, generalizing the Tutte-Beraha constants. - Jonathan Vos Post (jvospost3(AT)gmail.com), Nov 04 2007

Triangular sequence (gives the odd Tutte -Beraha constants as roots!)

{1},

{1, -1},

{1, -3, 1},

{1, -6, 5, -1},

{1, -10, 15, -7, 1},

{1, -15, 35, -28, 9, -1},

{1, -21, 70, -84, 45, -11, 1},

{1, -28, 126, -210, 165, -66, 13, -1},

{1, -36, 210, -462, 495, -286, 91, -15, 1},

{1, -45, 330, -924, 1287, -1001, 455, -120, 17, -1}

with(linalg): m:=(i, j)->min(i, j): M:=n->matrix(n, n, m): T:=(n, k)->coeff(charpoly(M(n), x), x, n-k): 1; for n from 1 to 11 do seq(T(n, k), k=0..n) od; # yields sequence in triangular form

An[d_] := MatrixPower[Table[Min[n, m], {n, 1, d}, {m, 1, d}], -1]; Join[{{1}}, Table[CoefficientList[CharacteristicPolynomial[An[d], x], x], {d, 1, 20}]]; Flatten[%]

Cf. A085478, A076756.

Cf. A085478, A076756.

Sequence in context: A121524 A103141 A085478 this_sequence A129818 A055898 A145904

Adjacent sequences: A123967 A123968 A123969 this_sequence A123971 A123972 A123973

sign,tabl

Gary Adamson and Roger Bagula (rlbagulatftn(AT)yahoo.com), Oct 29 2006

Edited by N. J. A. Sloane (njas(AT)research.att.com), Nov 29 2006