PARTITIONS OF NATURAL NUMBERS AND THEIR ORDERED REPRESENTATION FUNCTIONS

<jats:p> Let <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0004972725100890_inline1.png"/> <jats:tex-math>$\mathbb {N}$</jats:tex-math> </jats:alternatives> </jats:inline-formula> be the set of all nonnegative integers. For a set <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0004972725100890_inline2.png"/> <jats:tex-math>$A\subseteq \mathbb {N}$</jats:tex-math> </jats:alternatives> </jats:inline-formula> , let <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0004972725100890_inline3.png"/> <jats:tex-math>$R_2(A,n)$</jats:tex-math> </jats:alternatives> </jats:inline-formula> and <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0004972725100890_inline4.png"/> <jats:tex-math>$R_3(A,n)$</jats:tex-math> </jats:alternatives> </jats:inline-formula> be the number of solutions of the equation <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0004972725100890_inline5.png"/> <jats:tex-math>$n=a_1+a_2$</jats:tex-math> </jats:alternatives> </jats:inline-formula> with <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0004972725100890_inline6.png"/> <jats:tex-math>$a_1<a_2, a_1,a_2\in A$</jats:tex-math> </jats:alternatives> </jats:inline-formula> and with <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0004972725100890_inline7.png"/> <jats:tex-math>$a_1\le a_2, a_1,a_2\in A$</jats:tex-math> </jats:alternatives> </jats:inline-formula> , respectively. If <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0004972725100890_inline8.png"/> <jats:tex-math>$-N\le g\le N$</jats:tex-math> </jats:alternatives> </jats:inline-formula> , Yan [‘On the structure of partition which the difference of their representation function is a constant’, <jats:italic>Period. Math. Hungar.</jats:italic> <jats:bold>82</jats:bold> (2021), 149–152] showed that there is a set <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0004972725100890_inline9.png"/> <jats:tex-math>$A\subseteq \mathbb {N}$</jats:tex-math> </jats:alternatives> </jats:inline-formula> such that <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0004972725100890_inline10.png"/> <jats:tex-math>$R_i(A,n)-R_i(\mathbb {N}\setminus A,n)=g$</jats:tex-math> </jats:alternatives> </jats:inline-formula> for all integers <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0004972725100890_inline11.png"/> <jats:tex-math>$n\ge 2N-1$</jats:tex-math> </jats:alternatives> </jats:inline-formula> , where <jats:italic>N</jats:italic> is a positive integer. In this paper, we prove that if <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0004972725100890_inline12.png"/> <jats:tex-math>$g_1,g_2$</jats:tex-math> </jats:alternatives> </jats:inline-formula> are nonnegative integers with <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0004972725100890_inline13.png"/> <jats:tex-math>$g_1\neq g_2$</jats:tex-math> </jats:alternatives> </jats:inline-formula> , then there does not exist <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0004972725100890_inline14.png"/> <jats:tex-math>$A\subseteq \mathbb {N}$</jats:tex-math> </jats:alternatives> </jats:inline-formula> such that <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0004972725100890_inline15.png"/> <jats:tex-math>$R_i(A,2n)-R_i(\mathbb {N}\setminus A,2n)=g_1$</jats:tex-math> </jats:alternatives> </jats:inline-formula> and <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0004972725100890_inline16.png"/> <jats:tex-math>$R_i(A,2n+1)-R_i(\mathbb {N}\setminus A,2n+1)=g_2$</jats:tex-math> </jats:alternatives> </jats:inline-formula> for all sufficiently large integers <jats:italic>n</jats:italic> . </jats:p>