The Eulerian polynomials satisfy the recurrence relation $$x A_n(x) = \sum_{k=0}^{n} \binom{n}{k}(x-1)^{n-k} A_k(x).$$

This reminds me very much of $$0 = \sum_{k=0}^{b} \binom{b}{k}(-1)^{b-k}T_k$$ where $T_k$ counts the number of SSYT of shape $k^c$ with entries $1,2,\dots,n$ and $b=\binom{n}{c}.$

Are there any other combinatorial objects that satisfy a recurrence of length $n,$ where the summand is a binomial $\binom{n}{k}$ and with an alternating sign, in some sense? (The Eulerian polynomials yields and alternating sum for $x=0$, but the result is very unexiting.)

It feels like such recurrences arises quite naturally, and the "sign" part should make a counting argument with inclusion/exclusion easier. (The SSYT recuurence is such an example).