[{"key":1,"part":[{"a":"1","b":"a"},{"a":"23","b":"b"},{"a":"4","b":"c"},]},{"key":2,"part":[{"a":"1","b":"a"},{"a":"23","b":"b"}]}]

db.collection.aggregate([{"$match":{"$expr":{"$eq":["1234",{"$reduce":{"input":"$part","initialValue":"","in":{"$concat":["$$value","$$this.a"]}}}]}}}])