A note on the Bj\"{o}rner--Kalai theorem

In 1988, Bj\"{o}rner and Kalai used combinatorial shadow functions to characterize the maximal Betti sequence for a given $f$-vector and the minimal $f$-vector for a given Betti sequence. Their description of the maximal Betti sequence was expressed through a set of inequalities. In this paper, we introduce an error function $\delta_k$ associated with the combinatorial shadow functions and use it to sharpen these inequalities into exact equalities. As a corollary, we obtain an equivalent form of Bj\"{o}rner and Kalai's characterization of all possible pairs $(f,\beta)$ that can occur as the $f$-vector and Betti sequence of a simplicial complex. Moreover, combining our results with a previous result of Bj\"{o}rner in 2011, we derive a new number-theoretic inequality concerning the count of odd square-free integers with a specified number of prime factors.