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In the dyadic case the union of the Reverse H\"{o}lder classes, $RH_p^d$ isstrictly larger than the union of the Muckenhoupt classes $ A_p^d$. Weintroduce the $RH_1^d$ condition as a limiting case of the $RH_p^d$inequalities as $p$ tends to 1 and show the sharp bound on $RH_1^d$ constant ofthe weight $w$ in terms of its $A_\infty^d$ constant.  We also take a look at the summation conditions of the Buckley type for thedyadic Reverse H\"{o}lder and Muckenhoupt weights and deduce them from anintrinsic lemma which gives a summation representation of the bumped average ofa weight. Our lemmata also allow us to obtain summation conditions forcontinuous Reverse H\"{o}lder and Muckenhoupt classes of weights and bothcontinuous and dyadic weak Reverse H\"{o}lder classes. In particular, it showsthat a weight belongs to the class $RH_1$ if and only if it satisfies Buckley'sinequality. We also show that the constant in each summation inequality ofBuckley's type is comparable to the corresponding Muckenhoupt or ReverseH\"{o}lder constant.

Equivalent definitions of dyadic Muckenhoupt and Reverse H\"older  classes in terms of Carleson sequences, weak classes, and comparability of  dyadic $L\log L$ and $A_\infty$ constants