We consider the squares of the Rogers-Ramanujan functions and for each $S\in \{1,2\}$ we obtain a linear recurrence relation for the number of partitions of n into parts congruent to $\pm S\hspace{0.5em}\mathrm{mod}\hspace{0.5em}5$. In this context, we conjecture that for $1\le S<R,\hspace{0.5em}k\ge 1$, the theta series $$\frac{{(-1)}^k}{{(q^S,q^{R-S};q^R)}_{\infty }}\sum _{j=k}^{\infty }{(-1)}^j{q}^{j(j+1)R/2-jS}(1-q^{(2j+1)S})$$has non-negative coefficients. This improves a conjecture given by G. E. Andrews and M. Merca in 2012, which was proved independently three years later by A. J. Yee using combinatorial methods and R. Mao via partial theta functions. Combinatorial interpretations of this new conjecture give for each $S\in \{1,2,3,4\}$ an infinite family of linear homogeneous inequalities for the number of partitions of n into parts congruent to $\pm S\hspace{0.5em}\mathrm{mod}\hspace{0.5em}5$. Twenty identities involving Rogers-Ramanujan functions are experimentally discovered considering Jacobi’s triple product identity and Watson’s quintuple product identity.

Recently it was shown that the Fibonacci numbers can be expressed in terms of multinomial coefficients as sums over integer partitions into odd parts. In this paper, we introduce a similar representation for the Padovan numbers. As a corollary, we derive an infinite family of double inequalities.

The n th-order determinant of a Toeplitz-Hessenberg matrix is expressed as a sum over the integer partitions of n . Many combinatorial identities involving integer partitions and multinomial coefficients can be generated using this formula.