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for non-negative integers ''n'' and ''k'' where ''n'' ≥ ''k'' and with the initial condition ...

...olor="red">''ordered''</font> sum of <font color="red">''positive''</font> integers. A '''<math>k</math>-composition''' of <math>n</math> is a composition of < ...umber of solutions to <math>x_1+x_2+\cdots+x_k=n</math> in ''nonnegative'' integers. We call such a solution a '''weak <math>k</math>-composition''' of <math>n ...

...olor="red">''ordered''</font> sum of <font color="red">''positive''</font> integers. A '''<math>k</math>-composition''' of <math>n</math> is a composition of < ...umber of solutions to <math>x_1+x_2+\cdots+x_k=n</math> in ''nonnegative'' integers. We call such a solution a '''weak <math>k</math>-composition''' of <math>n ...

...olor="red">''ordered''</font> sum of <font color="red">''positive''</font> integers. A '''<math>k</math>-composition''' of <math>n</math> is a composition of < ...umber of solutions to <math>x_1+x_2+\cdots+x_k=n</math> in ''nonnegative'' integers. We call such a solution a '''weak <math>k</math>-composition''' of <math>n ...

...olor="red">''ordered''</font> sum of <font color="red">''positive''</font> integers. A '''<math>k</math>-composition''' of <math>n</math> is a composition of < ...umber of solutions to <math>x_1+x_2+\cdots+x_k=n</math> in ''nonnegative'' integers. We call such a solution a '''weak <math>k</math>-composition''' of <math>n ...

...olor="red">''ordered''</font> sum of <font color="red">''positive''</font> integers. A '''<math>k</math>-composition''' of <math>n</math> is a composition of < ...umber of solutions to <math>x_1+x_2+\cdots+x_k=n</math> in ''nonnegative'' integers. We call such a solution a '''weak <math>k</math>-composition''' of <math>n ...

...olor="red">''ordered''</font> sum of <font color="red">''positive''</font> integers. A '''<math>k</math>-composition''' of <math>n</math> is a composition of < ...umber of solutions to <math>x_1+x_2+\cdots+x_k=n</math> in ''nonnegative'' integers. We call such a solution a '''weak <math>k</math>-composition''' of <math>n ...

...olor="red">''ordered''</font> sum of <font color="red">''positive''</font> integers. A '''<math>k</math>-composition''' of <math>n</math> is a composition of < ...umber of solutions to <math>x_1+x_2+\cdots+x_k=n</math> in ''nonnegative'' integers. We call such a solution a '''weak <math>k</math>-composition''' of <math>n ...

...olor="red">''ordered''</font> sum of <font color="red">''positive''</font> integers. A '''<math>k</math>-composition''' of <math>n</math> is a composition of < ...umber of solutions to <math>x_1+x_2+\cdots+x_k=n</math> in ''nonnegative'' integers. We call such a solution a '''weak <math>k</math>-composition''' of <math>n ...

...olor="red">''ordered''</font> sum of <font color="red">''positive''</font> integers. A '''<math>k</math>-composition''' of <math>n</math> is a composition of < ...umber of solutions to <math>x_1+x_2+\cdots+x_k=n</math> in ''nonnegative'' integers. We call such a solution a '''weak <math>k</math>-composition''' of <math>n ...

:'''Instance''': <math>n</math> positive integers <math>x_1<x_2<\cdots <x_n</math>. ...

where <math>\mathbb{N}</math> is the set of all nonnegative integers. Compared to the LP for the max-flow problem, we just replace the last line Due to the Flow Integrality Theorem, when capacities are integers, there must be an integral flow whose value is maximum among all flows (int ...

...s: it guarantees that the median is always a list element (e.g., a list of integers will never have a fractional median), and it guarantees that the median exi ...

[[Category:Integers]] ...

...math>k</math>-compositions (the ''ordered'' sum of <math>k</math> positive integers). There are <math>{n-1\choose k-1}</math> many <math>k</math>-compositions ...integer <math>n</math>, let <math>\phi(n)</math> be the number of positive integers from <math>\{1,2,\ldots,n\}</math> that are relative prime to <math>n</math ...

...and <math>N</math> be a random variable taking values in the non-negative integers and independent of the <math>X_n</math> for all <math>n \ge 1</math>. Prove ...th> uniformly at random, where <math>n</math> and <math>q</math> are given integers with <math>q \ge 1</math>. Find the expected number of inversions in <math> ...

:'''Instance''': <math>n</math> positive integers <math>x_1<x_2<\cdots <x_n</math>. ...

:Let <math>k,\ell</math> be positive integers. Then there exists an integer <math>R(k,\ell)</math> satisfying: :Let <math>r, k_1,k_2,\ldots,k_r</math> be positive integers. Then there exists an integer <math>R(r;k_1,k_2,\ldots,k_r)</math> satisfyi ...

For any tuple <math>\mathbf{v}=(n_1,n_2,\ldots,n_m)</math> of nonnegative integers satisfying that <math>n_1+n_2+\cdots +n_m=n</math>, let <math>a_{\mathbf{v} Let <math>\mathbf{v}=(n_1,\ldots,n_m)</math> be a vector of nonnegative integers satisfying that <math>n_1+\cdots+n_m=n</math>, and let <math>a_{\mathbf{v}} ...

For any tuple <math>\mathbf{v}=(n_1,n_2,\ldots,n_m)</math> of nonnegative integers satisfying that <math>n_1+n_2+\cdots +n_m=n</math>, let <math>a_{\mathbf{v} Let <math>\mathbf{v}=(n_1,\ldots,n_m)</math> be a vector of nonnegative integers satisfying that <math>n_1+\cdots+n_m=n</math>, and let <math>a_{\mathbf{v}} ...

For any tuple <math>\mathbf{v}=(n_1,n_2,\ldots,n_m)</math> of nonnegative integers satisfying that <math>n_1+n_2+\cdots +n_m=n</math>, let <math>a_{\mathbf{v} Let <math>\mathbf{v}=(n_1,\ldots,n_m)</math> be a vector of nonnegative integers satisfying that <math>n_1+\cdots+n_m=n</math>, and let <math>a_{\mathbf{v}} ...

For any tuple <math>\mathbf{v}=(n_1,n_2,\ldots,n_m)</math> of nonnegative integers satisfying that <math>n_1+n_2+\cdots +n_m=n</math>, let <math>a_{\mathbf{v} Let <math>\mathbf{v}=(n_1,\ldots,n_m)</math> be a vector of nonnegative integers satisfying that <math>n_1+\cdots+n_m=n</math>, and let <math>a_{\mathbf{v}} ...

For any tuple <math>\mathbf{v}=(n_1,n_2,\ldots,n_m)</math> of nonnegative integers satisfying that <math>n_1+n_2+\cdots +n_m=n</math>, let <math>a_{\mathbf{v} Let <math>\mathbf{v}=(n_1,\ldots,n_m)</math> be a vector of nonnegative integers satisfying that <math>n_1+\cdots+n_m=n</math>, and let <math>a_{\mathbf{v}} ...

For any tuple <math>\mathbf{v}=(n_1,n_2,\ldots,n_m)</math> of nonnegative integers satisfying that <math>n_1+n_2+\cdots +n_m=n</math>, let <math>a_{\mathbf{v} Let <math>\mathbf{v}=(n_1,\ldots,n_m)</math> be a vector of nonnegative integers satisfying that <math>n_1+\cdots+n_m=n</math>, and let <math>a_{\mathbf{v}} ...

For any tuple <math>\mathbf{v}=(n_1,n_2,\ldots,n_m)</math> of nonnegative integers satisfying that <math>n_1+n_2+\cdots +n_m=n</math>, let <math>a_{\mathbf{v} Let <math>\mathbf{v}=(n_1,\ldots,n_m)</math> be a vector of nonnegative integers satisfying that <math>n_1+\cdots+n_m=n</math>, and let <math>a_{\mathbf{v}} ...

For any tuple <math>\mathbf{v}=(n_1,n_2,\ldots,n_m)</math> of nonnegative integers satisfying that <math>n_1+n_2+\cdots +n_m=n</math>, let <math>a_{\mathbf{v} Let <math>\mathbf{v}=(n_1,\ldots,n_m)</math> be a vector of nonnegative integers satisfying that <math>n_1+\cdots+n_m=n</math>, and let <math>a_{\mathbf{v}} ...

For any tuple <math>\mathbf{v}=(n_1,n_2,\ldots,n_m)</math> of nonnegative integers satisfying that <math>n_1+n_2+\cdots +n_m=n</math>, let <math>a_{\mathbf{v} Let <math>\mathbf{v}=(n_1,\ldots,n_m)</math> be a vector of nonnegative integers satisfying that <math>n_1+\cdots+n_m=n</math>, and let <math>a_{\mathbf{v}} ...

...integer <math>n</math>, let <math>\phi(n)</math> be the number of positive integers from <math>\{1,2,\ldots,n\}</math> that are relative prime to <math>n</math Let <math>U=\{1,2,\ldots,n\}</math> be the universe. The number of positive integers from <math>U</math> which is divisible by some <math>p_{i_1},p_{i_2},\ldots ...

...integer <math>n</math>, let <math>\phi(n)</math> be the number of positive integers from <math>\{1,2,\ldots,n\}</math> that are relative prime to <math>n</math Let <math>U=\{1,2,\ldots,n\}</math> be the universe. The number of positive integers from <math>U</math> which is divisible by some <math>p_{i_1},p_{i_2},\ldots ...

...integer <math>n</math>, let <math>\phi(n)</math> be the number of positive integers from <math>\{1,2,\ldots,n\}</math> that are relative prime to <math>n</math Let <math>U=\{1,2,\ldots,n\}</math> be the universe. The number of positive integers from <math>U</math> which is divisible by some <math>p_{i_1},p_{i_2},\ldots ...

...integer <math>n</math>, let <math>\phi(n)</math> be the number of positive integers from <math>\{1,2,\ldots,n\}</math> that are relative prime to <math>n</math Let <math>U=\{1,2,\ldots,n\}</math> be the universe. The number of positive integers from <math>U</math> which is divisible by some <math>p_{i_1},p_{i_2},\ldots ...

...integer <math>n</math>, let <math>\phi(n)</math> be the number of positive integers from <math>\{1,2,\ldots,n\}</math> that are relative prime to <math>n</math Let <math>U=\{1,2,\ldots,n\}</math> be the universe. The number of positive integers from <math>U</math> which is divisible by some <math>p_{i_1},p_{i_2},\ldots ...

...integer <math>n</math>, let <math>\phi(n)</math> be the number of positive integers from <math>\{1,2,\ldots,n\}</math> that are relative prime to <math>n</math Let <math>U=\{1,2,\ldots,n\}</math> be the universe. The number of positive integers from <math>U</math> which is divisible by some <math>p_{i_1},p_{i_2},\ldots ...

...roblem <math>f:\{0,1\}^*\to \mathbb{Z}^+</math> whose outputs are positive integers. ...roblem <math>f:\{0,1\}^*\to \mathbb{Z}^+</math> whose outputs are positive integers. ...

where <math>\mathbb{N}</math> is the set of all nonnegative integers. Compared to the LP for the max-flow problem, we just replace the last line Due to the Flow Integrality Theorem, when capacities are integers, there must be an integral flow whose value is maximum among all flows (int ...

...). For example, Briggs' first table contained the common logarithms of all integers in the range 1–1000, with a precision of 8 digits. As the function {{nowrap ...of logarithm tables: given a table listing log₁₀(''x'') for all integers ''x'' ranging from 1 to 1000, the logarithm of 3542 is approximated by ...

where <math>\mathbb{N}</math> is the set of all nonnegative integers. Compared to the LP for the max-flow problem, we just replace the last line Due to the Flow Integrality Theorem, when capacities are integers, there must be an integral flow whose value is maximum among all flows (int ...

where <math>\mathbb{N}</math> is the set of all nonnegative integers. Compared to the LP for the max-flow problem, we just replace the last line Due to the Flow Integrality Theorem, when capacities are integers, there must be an integral flow whose value is maximum among all flows (int ...

...d the smallest natural number. Natural numbers are always whole numbers ([[integers]]) and never less than zero. ...

...nstruction, a 2-universal hash function can be uniquely represented by two integers <math>a</math> and <math>b</math>, which can be stored in two entries (or j ...

:'''Input''': <math>n</math> positive integers <math>p_1,p_2,\ldots,p_n</math> and a positive integer <math>m</math>; :'''Input''': a set of <math>n</math> positive integers <math>S=\{x_1,x_2,\ldots,x_n\}</math>; ...

:'''Input''': <math>n</math> positive integers <math>p_1,p_2,\ldots,p_n</math> and a positive integer <math>m</math>; :'''Input''': a set of <math>n</math> positive integers <math>S=\{x_1,x_2,\ldots,x_n\}</math>; ...

:'''Input''': <math>n</math> positive integers <math>p_1,p_2,\ldots,p_n</math> and a positive integer <math>m</math>; :'''Input''': a set of <math>n</math> positive integers <math>S=\{x_1,x_2,\ldots,x_n\}</math>; ...

:'''Input''': <math>n</math> positive integers <math>p_1,p_2,\ldots,p_n</math> and a positive integer <math>m</math>; :'''Input''': a set of <math>n</math> positive integers <math>S=\{x_1,x_2,\ldots,x_n\}</math>; ...

:'''Input''': <math>n</math> positive integers <math>p_1,p_2,\ldots,p_n</math> and a positive integer <math>m</math>; :'''Input''': a set of <math>n</math> positive integers <math>S=\{x_1,x_2,\ldots,x_n\}</math>; ...