{5, 2},
{5, 1, 1},
{4, 3},
{4, 2, 1},
{4, 1, 1, 1},
{3, 3, 1},
{3, 2, 2},
{3, 2, 1, 1},
{3, 1, 1, 1, 1},
{2, 2, 2, 1},
{2, 2, 1, 1, 1},
{2, 1, 1, 1, 1, 1},
{1, 1, 1, 1, 1, 1, 1}

Length @ Catenate @ Map[Permutations] @ IntegerPartitions[7, 7, Range[5]]

permCount[list_] := Multinomial @@ Tally[list][[All,2]]

Total @ Map[permCount] @ IntegerPartitions[7, 7, Range[5]]

ClearAll["Global`*"]

 num[int_,max_]:=SeriesCoefficient[(1-x)/(1-2x+x^(max+1)),{x,0,int}];

num2=ParallelSum[(-1)^r Binomial[#1-#2*r-1, n-1] Binomial[n, r],{n,#1},{r,0,(#1 - n)/#2}]&;