We present a combinatorial model for the several kinds of polynomial sequences of binomial type and develop many of the theorems about them from this model. In the first section, we present a prefab model for the binomial formula and the generating-function theorem. In Sec. 2, we introduce the notion of V-graph and give examples of binomial prefabs of V-graphs. The umbral composition of V-graphs provides an interpretation of umbral composition of polynomial sequences in Secs. 3 and 5. Rota's interpretation of the Stirling numbers 'Of the first kind as sums 'Of the Mobius functi'On in the partiti'On lattice inspired 'Our model f'Or inverse sequences of binomial type in Sec. 4. Section 6 c'Ontains combinatorial pro'Ofs of several 'Operator-the'Oretic results. The acti'Ons of shift operators and delta operat'Ors are explained in set-the'Oretic terms. Finally, in Sec. 6 we give a model f'Or cross sequences and Sheffer sequences which is c'Onsistent with their decomp'Osition int'O sequences 'Of bin'Omial type.This provides an interpretati'On 'Of shift-invariant operators. Of course, all 'Of these interpretati'Ons require that the coefficients inv'Olved be integer and usually n'On-negative as well.

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Using the notion of convolution of binomial sequences it is possible to show that Sheffer sequences, cross sequences, and Steffensen sequences are only mild generalizations of ordinary sequences of binomial type.

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