The classical Pieri formula gives a combinatorial rule for decomposing the product of a Schur function and a complete homogeneous symmetric polynomial as a linear combination of Schur functions with integer coefficients. We give a Pieri rule for describing the product of an orthosymplectic character and an orthosymplectic character arising from a one-row partition. We establish that the orthosymplectic Pieri rule coincides with Sundaram's Pieri rule for symplectic characters and that orthosymplectic characters and symplectic characters obey the same product rule.
<p>Complete homogeneous symmetric polynomial has connections with binomial coefficient, composition, elementary symmetric polynomial, exponential function, falling factorial, generating series, odd prime and Stirling numbers of the second kind by different summations. Surprisingly the relations in the context are comparable in pairs. </p>
<p class="abstract">We count the number of occurrences of <em>t </em>as the summands<em> </em>(i) in the compositions of a positive integer <em>n</em> into <em>r</em> parts; and (ii) in all compositions of <em>n</em>; and subsequently obtain other results involving compositions. The initial counting further helps to solve the enumeration problems for complete homogeneous symmetric polynomial.</p>
An Orthosymplectic Pieri Rule