Consider a linear order on $[n] = \{1,\dots,n\}$ with dots between some of consecutive numbers in this order; examples $35.7.241.6$ or $62.175.43$ ($n = 7$). I call such a structure a dotted order on $[n]$ and denote it $(L,\lambda)$, where $L$ is a linear order and $\lambda$ is a subset of $\{1,\dots,n-1\}$. Given two orders $L$ and $M$ on $[n]$, denote $d(L,M)$ the choice of points showing the descents of $M$ with respect to $L$; example: $M = 235164$ has descents $23.5.1.64$ with respect to $L = 641253$, and $L$ has descents $64.1.25.3$ with respect to $M$. Let $A_n$ be the free abelian group on the dotted orders on $[n]$; its rank is clearly equal to $n!2^{n-1}$. Now consider the endomorphism $\phi_n$ of $A_n$ which sends each generator $(L, \lambda)$ to the sum of all $(M, \mu)$ such that $d(L,M) = \mu$ and $d(M,L) = \lambda$. Last year Nantel Bergeron raised the question to understand $\phi_n$ and to determine its kernel in particular.