Abstract
Abstract For an indifference graph G, we define a symmetric function of increasing spanning forests of G. We prove that this symmetric function satisfies certain linear relations, which are also satisfied by the chromatic quasisymmetric function and unicellular $\textrm {LLT}$ polynomials. As a consequence, we give a combinatorial interpretation of the coefficients of the $\textrm {LLT}$ polynomial in the elementary basis (up to a factor of a power of $(q-1)$ ), strengthening the description given in [4].
Highlights
We see that the coefficient of in the chromatic polynomial of counts the number of increasing spanning forests of with components. This interpretation holds for a larger class of graphs, namely, graphs that have a perfect elimination ordering
The chromatic polynomial of a graph admits a symmetric function generalization introduced by Stanley in [21]
Let Λ be the algebra of symmetric functions, and Λ ≔ Λ[ ]
Summary
Let Λ be the algebra of symmetric functions, and Λ ≔ Λ[ ]. We denote by ( ), h ( ), ( ), and ( ) the elementary, complete, Schur, and power sum symmetric functions. Before stating a proposition that relates ( ; ) with the complete homogeneous symmetric functions h ( ), we make a few definitions. We denote by ( ) the sum of the -weights of all domino tabloids of shape and type. Follow from item (1), just adapting a standard argument [17, page 109]. To prove item (6), we define ( ; ) = (1 − ) ( ; ) as in [17, Page 209]. Item (7) follows from [19, Theorem 4.13, item (d)]
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