Upon filing of a petition with the clerk of the board of county commissioners, the clerk shall publish a notice that a petition for the partition of an irrigation district has been filed, setting forth the first signature on the petition. The notice shall provide the time at which the petition will be presented to the board, which shall be during a regular meeting of the board or during a special meeting called for that purpose. The notice shall be published at least two (2) weeks prior to the day upon which the petition is to be presented in a newspaper of general circulation in the county. If any portion of the proposed partitioned district be within another county or counties, the notice shall also be published in a newspaper of general circulation in each of those counties.