Questions tagged [integer-partitions]
Use this tag for questions related to ways of writing a positive integer as a sum of positive integers.
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A generalized subset sum problem
I'm looking at a problem I believe is combinatorial.
Find all possible solutions $\mathbf{x}$ to:
$$\mathbf{a} = [a_1, a_2, ..., a_n], a_k \in \mathbb{N^+}$$
$$\mathbf{l} = [l_1, l_2, ..., l_n], l_k \...
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Name of such combinatorial numbers. [duplicate]
Let $k,n\in\mathbb{N}$. Let $N(k,n)$ denote the size of the finite set
$$\{(x_1,\cdots,x_k)\in\mathbb{N}^k:x_1+2x_2+\cdots+kx_k=n\}. $$
I feel it special and important. Do $N(k,n)$ have names? Is ...
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Partial Orders on Integer Partitions
My question is the following: An integer partition $\lambda$ can be represented as an integer sequence $(f_1,f_2,f_3, \cdots)$ where $f_i$ is the number of parts used in $\lambda$. For instance, $4 + ...
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Has anyone found a closed-form expression for the strict partition function?
More precisely, if anyone has found it, could they provide links, please? I have been trying to find such a solution and have not seen one.
For context, when I try to solve the strict partition ...
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Graded ring generated by finitely many homogeneous elements of positive degree has Veronese subring finitely generated in degree one
Let $S=\bigoplus_{k\ge 0}S_n$ be a graded ring which is generated over $S_0$ by some homogeneous elements $f_1,\dotsc, f_r$ of degrees $d_1,\dotsc, d_r\ge 1$, respectively. I want to show that there ...
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Congruences of partitions and Legendre symbol.
Let $(\frac{a}{p})$ denote the Legendre symbol, and let $\psi(q)=\sum_{n\geq 0} q^{n(n+1)/2}$.
We define Ramanujan's general theta function $f(a,b)$ for $\mid ab \mid <1$ as
$$
f(a,b)=\sum_{n=-\...
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"Factorization" of the solutions set of a system of linear diophantine equations over non-negative integers
Suppose we have a system of linear diophantine equations over non-negative integers:
$$
\left\lbrace\begin{aligned}
&Ax=b\\
&x\in \mathbb{Z}^n_{\geq0}
\end{aligned}\right.
$$
where $A$ is a ...
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irregularities in partition function modulo n
It is an open problem whether the partition function is even half the time. Inspired by this, I wrote some Sage/Python code to check how many times $p(n)$ hits each residue class:
...
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Show the partition function $Q(n,k)$, the number of partitions of $n$ into $k$ distinct parts, is periodic mod m
Let $Q(n,k)$ be the number of partitions of $n$ into $k$ distinct parts.
I want to show that for any $m \geq 1$ there exists $t \geq 1$ such that
$$Q(n+t,i)=Q(n,i) \mod m \quad \forall n>0 \forall ...
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A sum of multinomial coefficients over partitions of integer
I denote a partition of an integer $n$ by $\vec i = (i_1, i_2, \ldots)$ (with $i_1, i_2, \ldots \in \mathbb N$) and define it by
$$
\sum_{p\geq1} p i_p = n.
$$
I set
$$
|\vec i| = \sum_{p\geq1} i_p.
$$...
- 1,657
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A tagged partition interval subset relations problem.
The problem was taken from “Introduction to Real Analysis” by R. Bartle and D. Sherbert, Section 7.1, Exercise 4.
Let $\dot{\mathcal{P}}$ be a tagged partition of $[0,3]$.
(a) Show that the union $U_1$...
user1055322
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Find generating series on set of descending sequences, with weight function as taking sum of sequence
Given the set of all sequences of length k with descending (not strictly, so $3,3,2,1,0$ is allowed) terms of natural numbers (including $0$), $S_k$, and the weight function $w(x)$ as taking the sum ...
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Formula for number of monotonically decreasing sequences of non-negative integers of given length and sum?
What is a formula for number of monotonically decreasing sequences of non-negative integers of given length and sum? For instance, if length k=3 and sum n=5, then these are the 5 sequences that meet ...
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Integer partitions with summands bounded in size and number
This book says it's easy, but to me, it's not. :(
As for 'at most k summands', in terms of Combinatorics, by using MSET(),
$$ MSET_{\le k}(Positive Integer) = P^{1,2,3,...k}(z) = \prod_{m=1}^{k} \frac{...
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Generating Function for Modified Multinomial Coefficients
The multinomial coefficients can be used to expand expressions of the form ${\left( {{x_1} + {x_2} + {x_3} + ...} \right)^n}$ in the basis of monomial symmetric polynomials (MSP). For example,
$$\...
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