Crank

Ecology

Biology

Rank (graph theory)

Type (biology)

Mathematics

Cylinder

Congruence (geometry)

Geometry

Partition (number theory)

Combinatorics

Congruence relation

Modulo

International Journal of Number Theory

Let [Formula: see text] count the total number of parts among partitions of [Formula: see text] with rank congruent to [Formula: see text] modulo [Formula: see text] and let [Formula: see text] count the total appearances of ones among partitions of [Formula: see text] with crank congruent to [Formula: see text] modulo [Formula: see text]. We provide a list of over 70 congruences modulo [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] involving [Formula: see text] and [Formula: see text], which are known as congruences of Andrews–Beck type. Some recent conjectures of Chan, Mao and Osburn are also included in this list.

Weighted partition rank and crank moments. III. A list of Andrews–Beck type congruences modulo 5, 7, 11 and 13