Infrared spectra of the water clusters have been measured in the N2 + O2 matrix. The aggregation process of water in the matrix has been monitored by annealing the deposited samples up to 40 K and UV irradiation. The monomer, dimer, cyclic trimer and cyclic pentamer are found as water clusters in the matrix. For the hexamer, several structures such as chair, cage, prism, bag 1 and/or book 1 are likely to exist. By UV irradiation, the cyclic pentamer is predominantly formed from the monomer and dimer. On the other hand, by annealing the deposited sample, several hexamers are formed. The theoretical calculation for water clusters has revealed that the formation of one hydrogen bonding in a hydrogen-bonded chain cooperatively enhances or diminishes the strength of another hydrogen bond. Both proton donor (D) and acceptor (A) participating in a hydrogen-bonding pair DA are capable of forming hydrogen bonding with the other water molecules; D can additionally accept two protons and donate one proton, and A can additionally donate two protons and accept one proton. We have proposed the classification of hydrogen-bonding patterns considering the cooperativity, denoting as d'a'DAd''a'', where d and a are integers indicating the number of proton donors and acceptors to D (the single prime) and A (the double prime), respectively. Then, a magnitude given by MOH = -d' + a' + d'' - a'' has been introduced, which is very useful for connecting the hydrogen-bonding patterns to their OH wavenumbers. As a result, it is revealed that the OH stretching bands of water clusters are characterized by eight indicators (free and MOH = -2, -1, 0, 1, 2, 3 and 4). The classification proposed here is applicable to the OH band analysis for the hydrogen-bonded water and alcohols in a condensed phase.