Lang, Wolfdieter Combinatorial interpretation of generalized Stirling numbers. (English) Zbl 1165.05307 J. Integer Seq. 12, No. 3, Article ID 09.3.3, 26 p. (2009). Summary: A combinatorial interpretation of the earlier studied generalized Stirling numbers, emerging in a normal ordering problem and its inversion, is given. It involves unordered forests of certain types of labeled trees. Partition number arrays related to such forests are also presented. Cited in 10 Documents MSC: 05A15 Exact enumeration problems, generating functions 05A17 Combinatorial aspects of partitions of integers 05C05 Trees Keywords:increasing trees; enumeration; partition number arrays Software:OEIS PDFBibTeX XMLCite \textit{W. Lang}, J. Integer Seq. 12, No. 3, Article ID 09.3.3, 26 p. (2009; Zbl 1165.05307) Full Text: EuDML EMIS Online Encyclopedia of Integer Sequences: Triangle read by rows: the Bell transform of the triple factorial numbers A008544 without column 0. Numbers associated to partitions, used for combinatoric interpretation of Lah triangle numbers A105278; elementary Schur polynomials / functions. Partition number array, called M32(-1), related to A001497(n-1,m-1) = |S2(-1;n,m)| (generalized Stirling2 triangle). Partition number array, called M32hat(-1)= ’M32(-1)/M3’= ’A143171/A036040’, related to A001497(n-1,m-1)= |S2(-1;n,m)| (generalized Stirling triangle). Lower triangular array called S2hat(-1) related to partition number array A144269. Partition number array, called M32hat(-2)= ’M32(-2)/M3’= ’A143172/A036040’, related to A004747(n,m)= |S2(-2;n,m)| (generalized Stirling triangle). Lower triangular array called S2hat(-2) related to partition number array A144274. Partition number array, called M32hat(-3)= ’M32(-3)/M3’= ’A143173/A036040’, related to A000369(n,m)= |S2(-3;n,m)| (generalized Stirling triangle). Lower triangular array called S2hat(-3) related to partition number array A144279. Partition number array, called M31(-1), related to A049403(n,m) = S1(-1;n,m) (generalized Stirling triangle).