(a) The term "expenses of administration" means the total cost of administration of the department, excluding costs of providing services to municipalities and costs of administration of the health benefit plan, and excluding costs of special programs or activities of the department as may be determined by the president, subject to approval of the director of the budget, which do not serve generally all state departments and agencies under the jurisdiction of the department;
(b) The term "public authority" means a public authority or public benefit corporation under the jurisdiction of the department of civil service;
(c) The term "position" means an office or position, other than a seasonal or temporary position, in the classified service under the jurisdiction of the department of civil service. 2. A fractional share of the expenses of administration, as determined pursuant to subdivision three of this section, shall be charged against and payable from the following sources:
(a) If the salary or compensation of ten or more positions in any department or agency of the state is payable from a special or administrative fund, other than the state purposes fund or the local assistance fund or the capital construction fund of the general fund of the state, or an income fund of the state university, or the mental hygiene services fund, such fractional share shall be charged against and payable from such special or administrative fund;
(b) If the salary or compensation of ten or more positions in any department or agency of the state is payable from an appropriation assessed or to be collected or refunded pursuant to law, such fractional share shall be included in and payable from such appropriation;
(c) Such fractional share shall be payable by each public authority having ten or more positions. 3. For the purpose of computing the fractional share of the expenses of administration for any fiscal year to be payable from a source specified in subdivision two of this section, the numerator of such fraction shall be the number of positions the salary or compensation of which is payable from such source and the denominator shall be the total number of positions in the service of the state and public authorities. The numbers of positions shall be the numbers of positions existing as of October thirty-first in such fiscal year. 4. As soon as practicable after the close of the fiscal year ending March thirty-first, nineteen hundred sixty-seven, and thereafter as soon as practicable after the close of each fiscal year, the president shall determine the amount representing the share of expenses of administration for such fiscal year chargeable to each source specified in subdivision two of this section, and shall certify such amount to the appropriate department or agency of the state or public authority. Such sums shall be payable to the commissioner of taxation and finance at the beginning of the fiscal year following such certification.