Here is how I think it should go. First

$\mathrm{Sym}\left(V\oplus \wedge^2 V\right)\cong \mathrm{Sym}\left(V\right)\otimes \mathrm{Sym}\left(\wedge^2 V\right)$

Then we have

$\mathrm{Sym}\left(\wedge^2 V\right)\cong \sum\limits_{\lambda} \mathrm{Schur}_{\lambda}\left(V\right)$

where the sum is over partitions such that all parts of the conjugate partition are even. This came up in Symmetric tensor products of irreducible representations

The tensor product $\mathrm{Sym}\left(V\right)\otimes \mathrm{Schur}_{\lambda}\left(V\right)$ is known by Pieri's rule.

Now given a partition we take a maximal subdiagram such that every column has an even number of boxes. The complement is skew shape with at most one box in each column.

Further comment In response to the request for representation theoretic proofs of the results used see

MR1606831 (99b:20073) Goodman, Roe ; Wallach, Nolan R. Representations and invariants of the classical groups. Encyclopedia of Mathematics and its Applications, 68. Cambridge University Press, Cambridge, 1998. xvi+685 pp. ISBN: 0-521-58273-3; 0-521-66348-2

In particular see 9.2.2 Reciprocity rules for Pieri's rule and see 5.2.6 to see the decomposition of $\mathrm{Sym}\left(\wedge^2 V\right)$. The highest weight vectors are constructed using Pfaffians.

Here is how I think it should go. First

$\mathrm{Sym}\left(V\oplus \wedge^2 V\right)\cong \mathrm{Sym}\left(V\right)\otimes \mathrm{Sym}\left(\wedge^2 V\right)$

Then we have

$\mathrm{Sym}\left(\wedge^2 V\right)\cong \sum\limits_{\lambda} \mathrm{Schur}_{\lambda}\left(V\right)$

where the sum is over partitions such that all parts of the conjugate partition are even. This came up in Symmetric tensor products of irreducible representations

The tensor product $\mathrm{Sym}\left(V\right)\otimes \mathrm{Schur}_{\lambda}\left(V\right)$ is known by Pieri's rule.

Now given a partition we take a maximal subdiagram such that every column has an even number of boxes. The complement is skew shape with at most one box in each column.

Further comment In response to the request for representation theoretic proofs of the results used see

MR1606831 (99b:20073) Goodman, Roe ; Wallach, Nolan R. Representations and invariants of the classical groups. Encyclopedia of Mathematics and its Applications, 68. Cambridge University Press, Cambridge, 1998. xvi+685 pp. ISBN: 0-521-58273-3; 0-521-66348-2

In particular see 9.2.2 Reciprocity rules for Pieri's rule and see 5.2.6 to see the decomposition of $\mathrm{Sym}\left(\wedge^2 V\right)$. The highest weight vectors are constructed using Pfaffians.

Here is how I think it should go. First

$\mathrm{Sym}\left(V\oplus \wedge^2 V\right)\cong \mathrm{Sym}\left(V\right)\otimes \mathrm{Sym}\left(\wedge^2 V\right)$

Then we have

$\mathrm{Sym}\left(\wedge^2 V\right)\cong \sum\limits_{\lambda} \mathrm{Schur}_{\lambda}\left(V\right)$

where the sum is over partitions such that all parts of the conjugate partition are even. This came up in Symmetric tensor products of irreducible representations

The tensor product $\mathrm{Sym}\left(V\right)\otimes \mathrm{Schur}_{\lambda}\left(V\right)$ is known by Pieri's rule.

Now given a partition we take a maximal subdiagram such that every column has an even number of boxes. The complement is skew shape with at most one box in each column.

Here is how I think it should go. First

$\mathrm{Sym}\left(V\oplus \wedge^2 V\right)\cong \mathrm{Sym}\left(V\right)\otimes \mathrm{Sym}\left(\wedge^2 V\right)$

Then we have

$\mathrm{Sym}\left(\wedge^2 V\right)\cong \sum\limits_{\lambda} \mathrm{Schur}_{\lambda}\left(V\right)$

where the sum is over partitions such that all parts of the conjugate partition are even. This came up in Symmetric tensor products of irreducible representations

The tensor product $\mathrm{Sym}\left(V\right)\otimes \mathrm{Schur}_{\lambda}\left(V\right)$ is known by Pieri's rule.

Now given a partition we take a maximal subdiagram such that every column has an even number of boxes. The complement is skew shape with at most one box in each column.

Further comment In response to the request for representation theoretic proofs of the results used see

MR1606831 (99b:20073) Goodman, Roe ; Wallach, Nolan R. Representations and invariants of the classical groups. Encyclopedia of Mathematics and its Applications, 68. Cambridge University Press, Cambridge, 1998. xvi+685 pp. ISBN: 0-521-58273-3; 0-521-66348-2

In particular see 9.2.2 Reciprocity rules for Pieri's rule and see 5.2.6 to see the decomposition of $\mathrm{Sym}\left(\wedge^2 V\right)$. The highest weight vectors are constructed using Pfaffians.