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A157044 Triangle read by rows: T(n,k) = number of partitions of n into exactly k parts, each <= k.

%I A157044 #6 Jan 17 2013 02:54:42
%S A157044 1,0,1,0,1,1,0,1,1,1,0,0,2,1,1,0,0,2,2,1,1,0,0,2,3,2,1,1,0,0,1,4,3,2,
%T A157044 1,1,0,0,1,4,5,3,2,1,1,0,0,0,5,6,5,3,2,1,1,0,0,0,4,8,7,5,3,2,1,1,0,0,
%U A157044 0,4,9,10,7,5,3,2,1,1,0,0,0,3,11,12,11,7,5,3,2,1,1,0,0,0,2,11,16,14,11,7,5
%N A157044 Triangle read by rows: T(n,k) = number of partitions of n into exactly k parts, each <= k.
%C A157044 Also equals the number of partitions of n-1 into exactly k-1 parts, each <= k+1; equality reported by _Franklin T. Adams-Watters_.
%C A157044 Comment from _N. J. A. Sloane_: This is a special case of the fact that the number of partitions of a-c into exactly b-1 parts none exceeding c equals the number of partitions of a-b into exactly c-1 parts none exceeding b. See Andrews reference.
%C A157044 Row sums equal A064174; read backwards, rows approach the partition numbers A000041.
%D A157044 George E. Andrews, The Theory of Partitions, Addison-Wesley, Reading, Mass., 1976 (Theorem 1.5).
%e A157044 Table begins:
%e A157044 1
%e A157044 0,1
%e A157044 0,1,1
%e A157044 0,1,1,1
%e A157044 0,0,2,1,1
%e A157044 0,0,2,2,1,1
%e A157044 0,0,2,3,2,1,1
%e A157044 0,0,1,4,3,2,1,1
%e A157044 0,0,1,4,5,3,2,1,1
%e A157044 0,0,0,5,6,5,3,2,1,1
%e A157044 0,0,0,4,8,7,5,3,2,1,1
%e A157044 0,0,0,4,9,10,7,5,3,2,1,1
%e A157044 0,0,0,3,11,12,11,7,5,3,2,1,1
%e A157044 0,0,0,2,11,16,14,11,7,5,3,2,1,1
%e A157044 0,0,0,1,12,19,19,15,11,7,5,3,2,1,1
%e A157044 0,0,0,1,11,23,24,21,15,11,7,5,3,2,1,1
%e A157044 0,0,0,0,11,25,31,27,22,15,11,7,5,3,2,1,1
%e A157044 0,0,0,0,9,29,37,36,29,22,15,11,7,5,3,2,1,1
%e A157044 0,0,0,0,8,30,46,45,39,30,22,15,11,7,5,3,2,1,1
%e A157044 0,0,0,0,6,32,52,58,50,41,30,22,15,11,7,5,3,2,1,1
%e A157044 0,0,0,0,5,32,61,70,66,53,42,30,22,15,11,7,5,3,2,1,1
%e A157044 0,0,0,0,3,32,68,86,82,71,55,42,30,22,15,11,7,5,3,2,1,1
%e A157044 0,0,0,0,2,30,76,101,104,90,74,56,42,30,22,15,11,7,5,3,2,1,1
%e A157044 0,0,0,0,1,29,81,120,127,116,95,76,56,42,30,22,15,11,7,5,3,2,1,1
%e A157044 a(8,4)=4 since {2,2,2,2}, {3,2,2,1}, {3,3,1,1}, {4,2,1,1} are the partitions of 8 in exactly 4 parts <=4.
%t A157044 Table[T[n-1,k-1,k+1]-T[n-1,k-2,k+1],{n,20},{k,n}] with T[n,a,b] as defined in A047993.
%t A157044 <<DiscreteMath`Combinatorica`
%t A157044 partitionexact[ n_,m_ ]:=TransposePartition/@(Prepend[ #,m ]&/@Partitions[ n-m,m ] )
%t A157044 Table[Length@Select[partitionexact[n,k],Max[ # ]<=k&],{n,1,24},{k,n}]
%Y A157044 Cf. A000041, A157045, A157046, A047993
%K A157044 nonn,tabl
%O A157044 1,13
%A A157044 _Wouter Meeussen_, Feb 22 2009