Democratic Unionist leader Peter Robinson and Sinn Féin's Martin McGuinness are expected to consider the appointment of a new justice minister as part of talks about the allocation of ministries, which is based on the number of Assembly seats won.
The DUP topped Thursday's poll, winning 38 seats, while Sinn Féin is the largest nationalist party on 29, both groups increasing their representation since the last election.
The cross-community Alliance Party is expected to fill the Justice Minister's position under an agreement on policing and justice reached between the DUP and Sinn Féin during the last mandate.
The Alliance Party could be entitled to another executive seat after it increased its poll tally but the Ulster Unionists are expected to encourage re-elected independent former member David McClarty to return to the party, giving it enough Assembly strength to take the ministry from Alliance.
Stormont will sit for the first time on Thursday or Monday when ministerial portfolios will be chosen.
Mr McClarty refused to say what he planned to do next.
'I am elated at having achieved the victory, I will take time to enjoy the experience. At this stage I am not about to make any decisions whatsoever and I will take a day or two so that I can recover from the exertions of the campaign.'
The East Derry MLA, who left at Christmas following a dispute with his UUP local branch, said he would not endorse comments made by Ulster Unionist leader Tom Elliott, who branded Sinn Féin as 'scum' during an outburst at the Fermanagh and South Tyrone count.
Mr McGuinness said Mr Elliott had criticised a DUP and Sinn Féin carve up in the last mandate but this was no way for the UUP to build bridges.
Mr Robinson dedicated his poll-topping performance to Catholic Constable Ronan Kerr, 25, who died when a booby-trap bomb planted by dissident republicans exploded underneath his car in Omagh, Co Tyrone, last month.
He pledged to use his party's best Assembly election result to build a shared future for all in memory of the officer.
Mr McGuinness also committed his party to working in partnership at Stormont.
But while Sinn Féin claimed an extra seat, the Ulster Unionists and the nationalist SDLP both lost two Assembly spots, while Alliance gained a seat.
Mr Robinson said: 'I want to dedicate this victory to the memory of young Ronan Kerr, a young man who at 25 years of age was blown into eternity by the hands of evil men.'
Peter Robinson said 'he had a vision for Northern Ireland and he wanted to serve his community.'
Each of the 18 constituencies returned six Assembly members and after two long days of counts, the battle for the last seat proved crucial in determining the final make-up of the new Stormont administration.
In the end the DUP took 38 seats, Sinn Féin 29, UUP 16, SDLP 14 and Alliance eight.
The leader of the hardline Traditional Unionist Voice (TUV) Jim Allister took a seat in North Antrim after the ninth count and without reaching the quota.
He pledged to hound Sinn Féin and challenge the power-sharing government from within the Assembly.
The Greens' Stephen Agnew snatched a seat in North Down to keep his party at Stormont.
The election results were at times overshadowed by wrangles over delayed vote counting across Northern Ireland, but it was clear from an early stage that the dominance of the two big players in the unionist and nationalist blocs was to continue.
McGuinness criticises AV referendum
Earlier, the Northern Deputy First Minister Martin McGuinness hit out at the British Government for including the AV referendum with the assembly and council elections.
At the end of the Mid-Ulster count shortly before 10pm yesterday, Mr McGuinness said holding three elections on the one day complicated matters.
'Not just in relation to the count yesterday, and for that I think the British Government are culpable, particularly adding the AV referendum to two elections,' Mr McGuinness said.
Kenny congratulates Northern Assembly
Taoiseach Enda Kenny said the new Stormont Assembly has an unprecedented opportunity to consign Northern Ireland's troubled past further into history.
Mr Kenny congratulated the 108 successful candidates and praised voters for turning out.
'At the conclusion of this Assembly election, following the first-ever uninterrupted term of power-sharing government, we are reminded of the value of peaceful democratic politics and the need to constantly work to protect and renew our democratic institutions,' he said.
'But we also look ahead with greater confidence than ever before to a better, shared future on this island.
'The new Assembly begins its work from a very solid base.
'Although it faces significant challenges, which will require all parties to work closely together, it also has an unprecedented opportunity to put the past even further behind us and to bring about change for good.'
The Taoiseach added that the Government fully supported the Stormont Assembly.
'I reassure them that the Irish Government remains fully committed to working with them in a spirit of partnership and mutual respect,' he said.
