Questions tagged [co.combinatorics]
Enumerative combinatorics, graph theory, order theory, posets, matroids, designs and other discrete structures. It also includes algebraic, analytic and probabilistic combinatorics.
1,457
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More 3-connected cubic graphs with all 2-factors of same cycle type?
The setup is as in this question: Let $G$ be a 3-connected cubic graph. If all 2-factors of $G$ are isomorphic (as graphs), i.e. all have the same partition $\pi$ as cycle type, we'll say that $G$ is ...
4
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2
answers
616
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How to know if convex-hull of a set contains zero?
Let $(\lambda_1 , \cdots , \lambda_d) \vdash k$ be a partition of $k$ of length $d$. Is there any way to decide if $0 \in \text{Conv}\{(\underbrace{\alpha_1, \cdots, \alpha_1}_{\lambda_1}, \cdots , \...
4
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141
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Approximation of a convex shape in the $d$-dimensional Euclidean space for $d\gg 1$
We are given a convex shape $C$ lying inside the hypercube $[0,1]^d$ in the $d$-dimensional Euclidean space. Let the volume of $C$ be $\tfrac12$ (I guess nothing changes for any other fixed constant ...
4
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2
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488
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Geometry, Number Theory and Graph Theory of n-gon, permutation and graph labeling?
Given $n$ and $t$ lengths $ l_i, 1\leq l_1<l_2<\cdots<l_t\leq n-1$, of directed diagonals within an $n$-gon such that $l_1+\cdots+l_t\neq 0 \pmod n)$. Does it exist a directed path within ...
4
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2
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231
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Distribution of $0$-$1$ matrices
Consider $n\times n$ matrices with entries in $\{0,1\}$. The determinants of these ranges from $0$ to the Hadamard bound $\frac{(n+1)^{\frac{n+1}2}}{2^n}$. Assume $n$ is large enough.
What does the ...
4
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1
answer
678
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Seymour's second neighborhood conjecture
Does anyone out there know if Seymour's second neighborhood conjecture is still open? if not, I would appreciate any references.
4
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3
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327
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Diagonal asymptotics of integer compositions
A (weak) composition of a positive integer $n$ into $k$ parts is an ordered sequence of nonnegative integers $(a_1, a_2, \ldots, a_k)$ such that $ \sum_{i=1}^k a_i = n $. I am interested in the case ...
4
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1
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814
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Existence of triangle-free graphs for regular graphs of degree at most n/2
It is known that for triangle-free graphs, if they are $d$-regular, then $2d\leq n$, where $n$ is the number of vertices. In words, the degree is less than or equal to half the number of vertices (...
4
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0
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221
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Efficiently computing (plethysm-like?)substitutions of symmetric functions
This is a rather technical question, it arose in connection of some calculations that I need to have better grasp of the question Formal group law over $\mathbb{F}_p$ and my own older one What is ...
4
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1
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371
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Existence of a non-Eulerian atomistic lattice with this property on the Möbius function
Let $L$ be a finite lattice with least element $\hat{0}$, greatest element $\hat{1}$, and Möbius function $\mu$.
Question 1: What class of lattices the following property characterizes? $$\mu(\hat{0},...
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1
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Can we explicitly compute this "shift"-quantity over Boolean functions $u:\mathbb{F}_2^n\rightarrow \mathbb{F}_2$?
This question is a follow-up of this question.
Let $\mathbb{F}_2=\{0,1\}$ be the field with two elements, and suppose that $n$ is odd.
Question: Can we compute the exact minimum $$A:=
\min_{u:\mathbb{...
4
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1
answer
214
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Enumerating matrices function of ranks
Is there an expression/approximate expression for number of real matrices $M\in\{0,1\}^{n\times n}$ of rank $r\leq n$?
Is it known if $M$ is restricted to symmetric/skew-symmetric matrices?
Does ...
4
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1
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626
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Complete graph coloring with cycle restriction
The following question is probably open (It was posted on AoPS a long time ago, but no one has a solution)
We have a complete graph with $n$ vertices. Each edge is colored in one of $c$ colors such ...
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Determine binary function $f(x)$ by partial observation of $x$
Let $\boldsymbol{x} = (\boldsymbol{x}_1, \dots, \boldsymbol{x}_n)$ be a $n$-dimensional random vector on $\mathbb{R}$ (i.e. $\boldsymbol{x}$ is a random variable). Suppose we have a binary function
$f:...
4
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1
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839
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New edge coloring problem in graph theory
Let $G$ be a simple graph. Consider the following edge coloring:
We are allowed to use repetitive colors on some edges incident to a vertex such that the result does not contain a sequence of length $...
4
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1
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185
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On a double sum involving binomial coefficients
For natural $n$, let
\begin{equation}
p_n:=2^{1-n}\sum_{v=1}^l \binom l{(v+l)/2}1(v\equiv l)
\sum_{u=1-v}^{v-1}\binom k{(u+k)/2}1(u\equiv k), \tag{1}\label{1}
\end{equation}
where $k:=\lfloor(n+1)/...
4
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95
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Counting cycles after permuting within rows and columns
Consider a rectangular $p \times q$ array, labelled by the numbers $0, \ldots, pq - 1$ for convenience. Let $S_p$ and $S_q$ and $S_{pq}$ denote the symmetric groups. Take a family of permutations:
$...
4
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1
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532
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genus zero permutation and noncrossing partition
Question
Let $g$ to be an element of permutation group $S_n$, and $\tau = (1,2,3,\cdots,n)$ is the circular permutation. $g$ and $\tau g$ have $n+1$ cycles in total(fixed point is also a cycle), ...
4
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How big must the sumset $A+A$ be if $A$ satisfies no translation-invariant equations of low height?
Suppose $A$ is a finite subset of an abelian group. If there is no solution to $ma+nb=(m+n)c$ with $0\leq m,n\leq M$, can we bound $|A+A|$ from below? I am interested if one can obtain bounds much ...
4
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200
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Extract this constant term
Given a Laurent polynomial $F$ in the variables $\mathbf{t}=(t_1,\dots,t_n)$, let $CT_{\vec{\mathbf{t}}}\,F$ denote its constant term.
For example, $CT_{t_1,t_2}((8t_1-\frac1{3t_1t_2})(5t_1t_2+t_2^2+\...
4
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1
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924
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What is the minimum number of independent sets for a graph with fixed numbers of vertices and edges?
Fix integers $V$ and $E_{\text{max}}$, and consider graphs $G$ with $V$ vertices and at most $E_{\text{max}}$ edges. What is the best lower bound that one can give on the number of distinct ...
4
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1
answer
291
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Number of walks on integer lattice with self-edge at zero
Consider the graph with vertices $V=\mathbb Z$ and edges
$$E=\{(n,n+1):n\in\mathbb Z\}\cup\{(0,0)\},$$
that is,
the usual integer lattice with a self-edge at zero.
For some fixed parameters $a,b,n\in\...
4
votes
1
answer
288
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Collecting stones in n buckets
There are $n$ stones distributed in $n$ buckets (initially one stone in each bucket). At each step the content of each bucket is put in a random bucket, chosen independently out of a set of $n$ new ...
4
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195
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A matroid parity exchange property
As part of my research, I encountered the following problem. Let $M = (E,I)$ be a matroid and let $P = \{P_1,\ldots,P_n\}$ be a partition of $E$ into (disjoint) pairs. For $A \subseteq P$, we say that ...
4
votes
2
answers
656
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"half arithmetic progressions" in dense sets
Fix a positive real number d>0. Szemeredi's theorem implies that for every integer k, there exists an integer N(k,d) such that if A is a subset of the interval [1,N] with density greater than d >0, ...
4
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186
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Polynomials of growth for finite Heisenberg groups
Take a standard finite Heisenberg group with two standard generators and let's consider its growth polynomial - the polynomial which coefficients are equal to the sphere sizes.
For example for $H_3(Z/...
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214
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Optimal colorings
If $G=(V,E)$ is a graph, we call the smallest cardinal $\kappa$ such that there is a coloring map $c:V\to \kappa$ as the chromatic number of $G$ and denote it by $\chi(G)$.
For any coloring $c:V(G) \...
4
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0
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103
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Can we extend "every finite lattice is a sublattice of partitions of a finite set" to linear and/or finitary lattices?
Pudlák and Tůma https://link.springer.com/article/10.1007/BF02482893 proved that every finite lattice can be
embedded as a sublattice of the partition lattice of a finite set.
Can this be generalized ...
4
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2
answers
2k
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How to compute the rook polynomial of a Ferrers board?
Given a Ferrers board of shape $(b_1,\ldots,b_m)$, we define $r_k$ as number of ways to place $k$ non-attacking rooks (as in Chess). In section 2.4 of Stanley's Enumerative Combinatorics (vol. 1) it's ...
4
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104
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Weighted maximal number of disjoint singly-generated ideals in the divisibility poset for $\{1,2,\ldots,n\}$
In the mathoverflow question here the asymptotic growth of antichains in the divisibility poset ${\cal P}_n$ of the set of natural numbers $\{1,\ldots,n\}$ is considered. I have a somewhat dual ...
4
votes
2
answers
560
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Combinatorial interpretation for coefficients of reciprocal of power series
I've seen a number of combinatorial interpretations for the coefficients of the compositional inverse (aka reversion) of a power series. Is there a known combinatorial interpretation for the ...
4
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0
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808
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Number of arrangements that contain at least 1 path from top to bottom of 2D matrix
I have a $n\times n$ matrix of objects. $n'$ objects are black, and the rest $n^2-n'$ are white.
With that information, I can easily calculate the total number of black element arrangements that exist ...
4
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0
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252
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What is $\dim D^{\lambda}$ for the symmetric group?
What are the dimensions of the simple modules $D^{\lambda}=S^{\lambda}/S^{\lambda}\cap (S^{\lambda})^{\perp}$ for the modular representation theory of $S_n$, i.e. $\operatorname{char}(k)=p>0$?
I ...
4
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1
answer
534
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Diagonally-cyclic Steiner Latin squares
A Steiner triple system is a decomposition of $K_n$ into $K_3$, such as $S=\{013,026,045,124,156,235,346\}$. Steiner triple systems give rise to a Steiner Latin squares, such as $L$ below.
\[L=\left(...
4
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1
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223
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Asymptotics for 'generalized" Kasteleyn's formula
A follow up on an earlier MO question.
Kasteleyn's formula for the number of domino tilings of a $2n\times 2n$ square
$\prod_{j=1}^n\prod_{k=1}^n \left( 4\cos^2(\pi j/(2n+1))+4\cos^2(\pi k/(2n+1))\...
4
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0
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231
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How distributive are the bad Laver tables?
Suppose that $n\in\omega\setminus\{0\}$. Then define $(S_{n},*)$ to be the algebra where $S_{n}=\{1,...,n\}$ and $*$ is the unique operation on $S_{n}$ where
$n*x=x$
$x*1=x+1\,\text{mod}\, n$ and
if $...
4
votes
1
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366
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Variance of load in maximally loaded bin, if $m$ balls are thrown into $n$ bins
In the paper "Balls and bins a simple and tight analysis" by Raab and Steger, available here strong upper and lower bounds are proved about the number $M$ of balls in a maximally loaded bin when $m$ ...
4
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1
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Solving a Diophantine equation related to Algebraic Geometry, Steiner systems and $q$-binomials?
The short version of my question is:
1)For which positive integers $k, n$ is there a solution to the equation $$k(6k+1)=1+q+q^2+\cdots+q^n$$ with $q$ a prime power?
2) For which positive ...
4
votes
2
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320
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How many union-closed families in $\cal{P}(n)$?
A union-closed family $\cal{F} \subseteq \cal{P}([n])$ is a family such that for every $A,B \in \cal{F}$, $A \cup B \in \cal{F}$. Are there any reasonable approximations as to how many such families (...
4
votes
2
answers
388
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Congruence for complementary Bell numbers
The Bell numbers $B(n)$ can be given as a sum of the (signed) Stirling numbers of the second kind $S(n,k)$ as $B(n)=\sum_{k=0}^nS(n,k)$. There are also the so-called complementary Bell numbers defined ...
4
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2
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1k
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expected values over binomial distributions
In some works of economics/risk analysis etc., I have seen situations where people take the expected value of a function (such as a utility function/cost function) over a binomial distribution:
$$F(n)...
4
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1
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390
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Can $n$ circles on a plane generate $m$ intersection points where at least $k$ circles intersect?
Can $n$ circles on a plane generate $m$ intersection points where at least $k$ circles intersect?
For $k = 2$ the answer is obvious since we can always place circles so that every one of them ...
4
votes
1
answer
184
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"JigSaw Puzzle" on Set Family II
I was asked to post a different question following a wording error on my previous question, so here it is.
Given a set family $\mathcal{F}$ of $[n]$ (with certain additional properties), such that ...
4
votes
1
answer
324
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The existence of a specific kind of independent set in a connected graph satisfying the following property
Suppose $G$ is a connected finite graph satisfying that every edge $uv$ of $G$ belongs to a "triangle" $uvw$ such that $uv,uw\in E(G),\ vw\notin E(G)$ or $uv,vw\in E(G),\ uw\notin E(G)$(in other words,...
4
votes
1
answer
303
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Do graphs with large number of paths contain large chain minor?
Definition: A "$k$-chain" is a multi-graph obtained from a path of length $k$ by duplicating every edge.
Note that the number of paths between two endpoints of a $k$-chain is $2^k.$
Question: Let $G$...
4
votes
1
answer
291
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On the real and finite field rank of a $0/1$ matrix - I
Let $M\in\{-1,0,+1\}^{n\times n}$ be a matrix of rank $r$.
Consider the matrix $f(M)\in\{0,+1\}^{mn\times mn}$ where $0$ in $M$ is replaced by $m\times m$ all $0$ matrix, $+1$ in $M$ is replaced by $m\...
4
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0
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203
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Dimension of a certain space of symmetric functions: Part I
Let $s_{\lambda}$ denote the Schur polynomial associated to a partition $\lambda$. A partition $\lambda$ is called a $t$-core if none of its hook lengths are multiples of $t$.
QUESTION. Consider the ...
4
votes
2
answers
417
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Combination power elementary symmetric polynomial inequality
Combine my first previous question and second previous question with the Muirhead inequality. I have posed conjectures of two inequalities as follows:
Inequality 1: Let $n>2$ and $1 \le m \le n$...
4
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1
answer
292
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A Conjecture about the integral related to Chebyshev polynomial
I am interested in the following integral related to the Chebyshev polynomials
$$I_{n,m}:= \int_0^\pi \left(\frac {\sin nx}{\sin x}\right)^{m} dx,$$
where $n,m\in \mathbb{Z}^+.$
It is easy to see ...
4
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0
answers
573
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A combinatorial bound involving Stirling numbers of the second type
My previous question was solved in a very elegant way, hopefully this (seemingly more complicated) case is also easy for experts.
I need the inequality
$\Big(\prod^r_{i=1}p_i\Big)\sum^n_{j=0}(-1)^j\...