There are 649 seats in the houde of commons. The speaker is neutral and doesn’t vote and 5 Sinn Fein MP’s don’t take their seats.That leaves 343 that can vote in a partisan way. The target would, therefore, seem to be 322 not 323.
Labour and Liberals together have 315 (7 short) but the Nationalist Parties have 9 votes between them and both seem anxious to vote with Labour against the Tory programme.
Added to this the 3 SDLP MP’s usually take the Labour whip and given that the Greens and the Alliance Party in Northern Ireland are also left-leaning parties, a progressive coalition would not be too uncomfortable.