With the mayor’s race over, the jockeying begins in another contest for a top L.A. City Hall post: the presidency of the City Council.

Current council president Alex Padilla is looking for his third two-year term, but he is being challenged by Hollywood area Councilman Eric Garcetti.

A council president can have significant influence in shaping the chamber’s agenda. That includes appointing councilmembers to committees, being acting mayor when the mayor is out of town (as Padilla did when James Hahn was stranded in Washington after the terrorist attacks), and making commission appointments if the mayor doesn’t fill them after a certain period of time.

Padilla has said privately that he has eight votes necessary to secure a majority of the 15-member council. However, Garcetti supporters say their man can peel off a couple of those to get to eight himself.

Those supporters note residual bad feeling on how Padilla presided over the bitter debate on whether to place on the ballot a half-cent sales tax hike for additional cops. Padilla sided with Antonio Villaraigosa, Greig Smith, Jack Weiss and Dennis Zine against the proposal, which needed 10 votes to pass.

Last week’s elections add more uncertainty, given that Villaraigosa will be off the council when the vote occurs this summer. That means whoever wins the presidency must get the support of seven of the remaining 13 councilmembers, instead of 7 out of 14 members, leaving less wriggle-room.

Then there’s the election of former cable television executive and talk show host Bill Rosendahl to the council. Garcetti supported Rosendahl in the council race, but Padilla supported Rosendahl’s opponent, Flora Gil Krisiloff. The expectation is that Rosendahl would initially side with Garcetti.

The focus will be on locking up swing votes, including Janice Hahn, Jan Perry, Jack Weiss and possibly Bernard Parks. (Parks proved to be the swing vote two years ago when he sided with Padilla against Wendy Greuel.)

Here is where the jockeying will be most intense. In exchange for their votes, councilmembers seek key committee assignments. Parks, for example, received the chairmanship of the powerful budget committee in exchange for his vote for Padilla.

Another question is where labor will side. As a self-described progressive, Garcetti has frequently aligned himself with labor and is now pushing for passage of a labor-backed ordinance that requires developers to set aside housing units as affordable. Padilla, though, has been a reliable supporter of public employee unions on contracts.

There’s also the question of which councilmember’s bid mayor-elect Villaraigosa will support, if any. While he won’t have a direct vote, Villaraigosa can indicate who he would rather work with as council president.

*The full column is available in the May 23 edition of the Los Angeles Business Journal.