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Dragan Stevanovic
Dragan Stevanovic
  • 13
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Counting certain walks in threshold graphs, I came to the following independent problem. Assume that $x_1,\dots,x_a$ are independent variables and for $k\geq 2$ let $$ P_k(x_1,\dots,x_a)=\sum_{(i_1,\dots,i_k)\in\{1,\dots,a\}^k} x_{i_1} x_{\min(i_1,i_2)} x_{\min(i_2,i_3)}\cdots x_{\min(i_{k-1},i_k)} x_{i_k}. $$ Can you find the coefficient of the term $x_{j_1}x_{j_2}\cdots x_{j_{k+1}}$ for any given $1\leq j_1\leq j_2\leq\dots\leq j_{k+1}\leq a$, or better yet, a closed-form formula for $P_k(x_1,\dots,x_a)$? If this problem appeared elsewhere earlier, a reference is more than welcome.

Counting certain walks in threshold graphs, I came to the following independent problem. Assume that $x_1,\dots,x_a$ are independent variables and let $$ P_k(x_1,\dots,x_a)=\sum_{(i_1,\dots,i_k)\in\{1,\dots,a\}^k} x_{i_1} x_{\min(i_1,i_2)} x_{\min(i_2,i_3)}\cdots x_{\min(i_{k-1},i_k)} x_{i_k}. $$ Can you find the coefficient of the term $x_{j_1}x_{j_2}\cdots x_{j_{k+1}}$ for any given $1\leq j_1\leq j_2\leq\dots\leq j_{k+1}\leq a$, or better yet, a closed-form formula for $P_k(x_1,\dots,x_a)$? If this problem appeared elsewhere earlier, a reference is more than welcome.

Counting certain walks in threshold graphs, I came to the following independent problem. Assume that $x_1,\dots,x_a$ are independent variables and for $k\geq 2$ let $$ P_k(x_1,\dots,x_a)=\sum_{(i_1,\dots,i_k)\in\{1,\dots,a\}^k} x_{i_1} x_{\min(i_1,i_2)} x_{\min(i_2,i_3)}\cdots x_{\min(i_{k-1},i_k)} x_{i_k}. $$ Can you find the coefficient of the term $x_{j_1}x_{j_2}\cdots x_{j_{k+1}}$ for any given $1\leq j_1\leq j_2\leq\dots\leq j_{k+1}\leq a$, or better yet, a closed-form formula for $P_k(x_1,\dots,x_a)$? If this problem appeared elsewhere earlier, a reference is more than welcome.

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Dragan Stevanovic
Dragan Stevanovic
  • 13
  • 3

Counting certain walks in threshold graphs, I came to the following independent problem. Assume that $x_1,\dots,x_a$ are independent variables and let $$ P_k(x_1,\dots,x_a)=\sum_{(i_1,\dots,i_k)\in\{1,\dots,a\}^k} x_{i_1} x_{\min(i_1,i_2)} x_{\min(i_2,i_3)}\cdots x_{\min(i_{k-1},i_k} x_{i_k}. $$$$ P_k(x_1,\dots,x_a)=\sum_{(i_1,\dots,i_k)\in\{1,\dots,a\}^k} x_{i_1} x_{\min(i_1,i_2)} x_{\min(i_2,i_3)}\cdots x_{\min(i_{k-1},i_k)} x_{i_k}. $$ Can you find the coefficient of the term $x_{j_1}x_{j_2}\cdots x_{j_k}$$x_{j_1}x_{j_2}\cdots x_{j_{k+1}}$ for any given $1\leq j_1\leq j_2\leq\dots\leq j_k\leq a$$1\leq j_1\leq j_2\leq\dots\leq j_{k+1}\leq a$, or better yet, a closed-form formula for $P_k(x_1,\dots,x_a)$? If this problem appeared elsewhere earlier, a reference is more than welcome.

Counting certain walks in threshold graphs, I came to the following independent problem. Assume that $x_1,\dots,x_a$ are independent variables and let $$ P_k(x_1,\dots,x_a)=\sum_{(i_1,\dots,i_k)\in\{1,\dots,a\}^k} x_{i_1} x_{\min(i_1,i_2)} x_{\min(i_2,i_3)}\cdots x_{\min(i_{k-1},i_k} x_{i_k}. $$ Can you find the coefficient of the term $x_{j_1}x_{j_2}\cdots x_{j_k}$ for any given $1\leq j_1\leq j_2\leq\dots\leq j_k\leq a$, or better yet, a closed-form formula for $P_k(x_1,\dots,x_a)$? If this problem appeared elsewhere earlier, a reference is more than welcome.

Counting certain walks in threshold graphs, I came to the following independent problem. Assume that $x_1,\dots,x_a$ are independent variables and let $$ P_k(x_1,\dots,x_a)=\sum_{(i_1,\dots,i_k)\in\{1,\dots,a\}^k} x_{i_1} x_{\min(i_1,i_2)} x_{\min(i_2,i_3)}\cdots x_{\min(i_{k-1},i_k)} x_{i_k}. $$ Can you find the coefficient of the term $x_{j_1}x_{j_2}\cdots x_{j_{k+1}}$ for any given $1\leq j_1\leq j_2\leq\dots\leq j_{k+1}\leq a$, or better yet, a closed-form formula for $P_k(x_1,\dots,x_a)$? If this problem appeared elsewhere earlier, a reference is more than welcome.

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Dragan Stevanovic
Dragan Stevanovic
  • 13
  • 3

Closed-form formula for a multivariate polynomial

Counting certain walks in threshold graphs, I came to the following independent problem. Assume that $x_1,\dots,x_a$ are independent variables and let $$ P_k(x_1,\dots,x_a)=\sum_{(i_1,\dots,i_k)\in\{1,\dots,a\}^k} x_{i_1} x_{\min(i_1,i_2)} x_{\min(i_2,i_3)}\cdots x_{\min(i_{k-1},i_k} x_{i_k}. $$ Can you find the coefficient of the term $x_{j_1}x_{j_2}\cdots x_{j_k}$ for any given $1\leq j_1\leq j_2\leq\dots\leq j_k\leq a$, or better yet, a closed-form formula for $P_k(x_1,\dots,x_a)$? If this problem appeared elsewhere earlier, a reference is more than welcome.