Questions tagged [partitions]
The partitions tag has no usage guidance.
391
questions
3
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How to find the right path of integration to get the asymptotic partition formula
I am trying to understand how the asymptotic partition formula $p(n) \sim \frac{e^{\pi\sqrt{\frac{2n}{3}}}}{4n\sqrt3} $ was derived for a project and have been reading and following many papers.
I am ...
3
votes
0
answers
180
views
+50
Number of partitions of set restricted by sum of square of part size
Let $p_1^{a_1}p_2^{a_2}\cdots$ denotes the integer partition of $n$, i.e. $a_1p_1+a_2p_2+\cdots=n$. Or equivalently $m_1+m_2+\cdots=n$. It is known that the number of partitions of set $\{x_1,x_2,\...
- 261
0
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0
answers
152
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Sum of square of parts, and sum of binomials over integer partition
Let $n$ be positive integer. Consider its integer partitions denoting as $(m_1,\cdots,m_k)$, where $m_1+\cdots+m_k=n$ and the order does not matter. We ignore the case of $(m_1,\cdots,m_k)=n$.
I am ...
- 261
4
votes
0
answers
148
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Sum $f(n_1,n_2,\ldots,n_k) 1^{n_1} 2^{n_2} \ldots k^{n_k}$ over partitions
Use the notation $(n_1,n_2,\ldots,n_k) \vdash n$ to denote that $(n_1,n_2,\ldots,n_k)$ is a partition of the positive integer $n$, that is, $n_1+n_2+\ldots+n_k = n$ and $n_1 \ge n_2 \ge \ldots \ge n_k ...
- 261
3
votes
2
answers
147
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Finding an inclusion-based path through 2-part set partitions
Given $S = \{1, 2, \ldots, n\}$, consider partitions of $S$ of the form $(R, R')$ where $R \subset S$ and $R'$ is $S \setminus R$, the complement of $R$ in $S$. The goal is to list 2-part partitions ...
- 3,986
1
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0
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93
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Pretty simple recursion for the A290383
Let $a(n)$ be A290383 i.e. number of set partitions of $[n]$ such that the smallest element of each block is odd. Here
$$
a(n)=b(n,0,0)
$$
where
$$
b(n,m,t)=\sum\limits_{j=1}^{m-t+1}b(n-1,\max(m,j),1-...
- 2,790
2
votes
1
answer
130
views
Number of partitions of an integer subject to some restrictions
Given a multiset $S$ of integers and an integer $n$. The size of $S$ is $n$ and each of the elements of $S$ lie within the range $1$ to $n-1$. Give a tight upper bound (in terms of $n$) on the number ...
- 21
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2
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269
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Number of partitions of $n$ and number of different integers in 1-avoiding partitions
Consider the number of integer partitions of $n$, usually denoted by $p(n)$ and generated by
$$\sum_{n\geq0}p(n)x^n=\prod_{k\geq1}\frac1{1-x^k}.$$
I have encountered an interesting enumeration.
Take ...
- 41k
2
votes
0
answers
68
views
Recursion for the number of partitions of $m^n-1$ into powers of $m$
Let $a(n,m)$ be the number of partitions of $m^n-1$ into powers of $m$. In other words,
$$a(n,m)=[z^{m^n-1}] \prod\limits_{k\geqslant 0} \frac{1}{1-z^{m^k}}$$
Let
$$
R(n,m,q)=\sum\limits_{j=0}^{m(q+1)-...
- 2,790
2
votes
0
answers
68
views
Skewed plane partition with only row fillings reversed
The number of plane partitions in a bounded box is well-studied and dates back to MacMahon, at the start of this paper by Sam Hopkins and Tri Lai, p9, they summarized current results on the ...
- 21
1
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0
answers
83
views
Reference for modularity of the Andrews–Gordon–Rogers–Ramanujan identities?
The right-hand side of the identity https://mathworld.wolfram.com/Andrews-GordonIdentity.html is a $q$-series $\frac{(q^i,q^{2k+1-i},q^{2k+1};q^k)_\infty}{(q;q)_\infty}$; is there a reference of its ...
- 390
8
votes
1
answer
239
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What is the Möbius function for the lattice of partial partitions?
Let $n$ be a positive integer. Let $P$ be the set of partitions of subsets of
$\{ 1, 2, \dotsc, n \}$ (so, for example, when $n=2$, the set $P$ contains $\emptyset$, $\{ \{1 \} \}$, $\{ \{2 \} \}$, $\{...
- 397
3
votes
1
answer
194
views
Seeking for a combinatorial argument for partition identities
Given an integer partition $\lambda$, introduce the following quantities:
\begin{align*}
c(\lambda)&=\sum_{i\geq1}\left\lceil\frac{\lambda_i}2\right\rceil, \qquad c_o(\lambda)=\sum_{i\geq1}\left\...
- 41k
0
votes
0
answers
169
views
Are the numbers $\sum_{n=1}^\infty\frac1{p(n)}$ and $\sum_{n=1}^\infty\frac1{q(n)}$ transcendental?
For each positive integer $n$, let $p(n)$ be the number of partitions of $n$ (i.e., the number of ways to write $n$ as a sum of positive integers), and let $q(n)$ be the number of strict partitions of ...
- 13.9k
2
votes
0
answers
219
views
Ramanujan's theta functions and hook lengths?
Given an integer partition $\lambda\vdash n$ of $n$, one may associate a Young diagram $Y(\lambda)$ to it followed by a computation of hook length $h_{\square}$ for each cell $\square=(i,j)$ in $Y(\...
- 41k
6
votes
2
answers
415
views
Plane partitions as sums of determinants
Consider the Vandermonde's determinant computed by
$$V(x_1,\dots,x_m):=\det(x_j^{i-1})_{i,j=1}^m=\prod_{1\leq i<j\leq m}(x_i-x_j).$$
The number of plane partitions in an $n\times m\times m$ box (...
- 41k
4
votes
0
answers
168
views
Olympiad problem relevant to $(a,b)$-feasible pair
Recently, a mathematical olympiad problem is proposed as follows:
Let $G$ be a graph with $|V| = 100$ and $\delta(G) \geqslant 10$. Prove that there is an integer $0 \leqslant k \leqslant 5$, such ...
3
votes
0
answers
108
views
Intersection numbers of moduli spaces and noncrossing partitions
The coefficients of the monomials $u_1^{e_1}u_2^{e_2} \ldots u_n^{e_n}$ of the partition polynomials (ParPs) $[M=M1]$ on pg. 831 of The Handbook of Mathematical Functions by Abramowitz and Stegun are ...
- 9,377
0
votes
0
answers
105
views
The number of partitions of a positive integer allowing at most r repetitions of any part
Let $q_r(n)$ be the number of partitions of the positive integer $n$ allowing at most $r$ repetitions of any of the parts. (For $r=1$ this is just the usual number of partitions of $n$ into distinct ...
- 97
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votes
0
answers
108
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Validating a result on evaluating Jack polynomials
I am currently working through the following paper:
Lapointe L., Lascoux A., Morse J.
Determinantal Expression and Recursion for Jack Polynomials
Electron. J. Combin. 7 (2000), Notes 1.
DOI: 10.37236/...
4
votes
1
answer
193
views
Fast computation of the partition function modulo a prime
Rademacher’s formula for the partition function allows fast computation using high precision arithmetic, but requiring a lot of memory. Here is an example computation of $p(10^{20})$ by Fredrik ...
- 1,558
1
vote
0
answers
132
views
A representation problem involving strict partition numbers
For each positive integer $n$, let $q(n)$ denote the number of ways to write $n$ as a sum of distinct positive integers. We call those $q(n)\ (n=1,2,3,\ldots)$ strict partition numbers.
The sequence $...
- 13.9k
0
votes
0
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107
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Is it express in terms of Schur Q-function?
Consider next integral
\begin{eqnarray}
Z \ = \ h^{- N N_f} \ \int\limits_{SU(N)} \ dU \ \prod_{n=1}^{N} \
\det \left ( 1 + h U \right )^{ N_f} \
\left ( 1 + h U^{\dagger} \right )^{ N_f} \ = \sum_{...
2
votes
1
answer
241
views
Optimal algorithm for a "round robin" doubles tournament?
Side note: so far neither Bard nor ChatGPT has managed to do this correctly, even when I show the errors.
I have 4N players ( N = 4 or N = 5 suffices) and want to set up three rounds of play. In each ...
- 133
0
votes
0
answers
96
views
I search representation in terms of Schur Q-function
Consider next sum
$$
Z_0^{N, N_f} = \sum_{r=0}^{N N_f} \sum_{\lambda \vdash r }s_{\lambda}(1^{N_f})
s_{\lambda} (1^{N_f})
= \det_{1\le i, j \le N} \ \binom{2 N_f}{N_f-i +j} = s_{N^{N_f}} \left(...
3
votes
2
answers
225
views
Proof of an asymptotic formula by Tricomi
Firs of all I ask my question, then I explain how this question arises in my mind and lastly what I tried to solve it.
QUESTION:
Let $P_{n,N}(k)$ be the number of composition of an integer $k$ in $n$ ...
- 177
1
vote
1
answer
116
views
Quantitative version of Lebesgue points theorem
Let $A \subset [0,1]^n$ with $A$ measurable and such that $\mathcal{L}^n (A)= \delta >0$, and consider a partition of $[0,1]^n$ in $\epsilon$-cubes (i.e. cubes of side $\epsilon)$. For $\epsilon \...
- 383
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answers
116
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How much choice we can get from this partition principle?
For every set $X$ there cannot be a partition on $X$ of a larger size than the set $\iota``X$ of all singleton subsets of $X$. Formally: $$\begin{align} \forall X \forall P: P \text { is a partition ...
- 7,891
2
votes
0
answers
199
views
Toric decomposition of multipartitions
Fix $k \in \mathbb Z_{>0}$. By a $k$-multipartition $\lambda=(\lambda_1,\dots,\lambda_k)$ of $N$, I mean that each $\lambda_i$ is a partition of some $N_i$ and $\sum N_i = N$.
Let's call $\lambda$ ...
- 101
1
vote
1
answer
185
views
hook length formula for plane partitions
The hook length formula give a simple product expression for the number of standard Young tableaux of a given shape $\lambda$, where $\lambda$ is an integer partition, or equivalently, the number of ...
- 528
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votes
1
answer
235
views
Identity involving Stirling number of the second kind
I'm looking for a citable reference for the following identity involving the Stirling numbers of the second kind $S(n, k)$ stated in Equation (27): For $n \geq 2$,
$$
\sum_{m=1}^n S(n, m) (-1)^m (m-1)!...
- 58
1
vote
1
answer
135
views
Largest part and length of a partition in play
If $\lambda=(\lambda_1\geq\lambda_2\geq\dots\geq\lambda_k)\vdash n$ is an integer partition of $n$ then $\lambda_1$ is its largest part and $k$ is its length, $\ell(\lambda)$.
Define the statistic $...
- 41k
5
votes
1
answer
176
views
What is the name for an integer partition with bounded multiplicities?
Is there a standard name for integer partitions $\lambda \in (\mathbb{Z}_{\geq 0})^n$, $\lambda_i \geq \lambda_{i+1}$, with multiplicities at most $k$, i.e. $\lambda_i > \lambda_{i+k}$ for all $i$?
...
- 1,641
1
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0
answers
70
views
Shuffling $\omega$ fairly for a fixed partition
Let ${\frak P}\subseteq {\cal P}(\omega)$ be a partition such that every block $B\in {\frak P}$ contains at least two integers.
Is there a countable set ${\cal F}$ of bijections $\varphi:\omega\to\...
- 43.6k
1
vote
1
answer
161
views
Partitioning $\mathbb R$ into sets such that no mutual points have distance $1$ [closed]
I was trying to partition $\mathbb R$ into two sets $A, B$ such that for all $a\in A, b\in B$ we have $|a-b|\neq 1$. An obvious way to do it is to take $\mathbb Z$ and ${\mathbb R}\setminus {\mathbb Z}...
- 43.6k
1
vote
0
answers
86
views
Intersection of schubert varieties
Let $L_1$ and $L_2$ $\in$ $\mathbb{P}^4$ be two planes that intersect in exactly one point $Q$. Let $P_1 \in L_1$, $P_2 \in L_2$ points, such that $P_1 \neq Q \neq P_2$. Using the duality theorem, ...
- 11
1
vote
2
answers
532
views
Terminology for a bijection from a set to itself
A current project uses bijections from a set to itself. (The set is the integer compositions of $n$, i.e., "ordered partitions of $n$," but that doesn't seem pertinent to the question.) Is ...
- 3,986
4
votes
1
answer
139
views
Prime numbers and number of partitions of $n$ into distinct parts with boundary size $2$
Let $a(n)$ be A227559, i.e., number of partitions of $n$ into distinct parts with boundary size $2$. Be careful here: offset is $3$.
I conjecture that $a(4n+2)=2n+1$ for $n>0$ if and only if $2n+1$ ...
- 2,790
5
votes
0
answers
105
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Sum of Schur functions associated to self-conjugate partitions
The $\tau$-function $H^\circ \big(t ;\vec{x} \big)$ associated with counting simple Hurwitz numbers is the formal power series
\begin{equation}
(\dagger) \quad H^\circ \big(t ;\vec{x} \big) \, =
\,
\...
- 1,767
0
votes
0
answers
125
views
Truncated circle method and partitions
Let $p(n)$ be the unrestricted partition function. Then it is known that its generating function $F(z)$ is expressible as an infinite product
$$
F(z)=1+\sum_{n\ge1}p(n)z^n=\prod_{k\ge1}(1-z^k)^{-1}.
$$...
- 1,285
1
vote
0
answers
68
views
Conjecture on numbers $k$ having only one partition into parts with same binary weight as a binary weight of $k$
Let $\operatorname{tr}(n)$ be A007814, number of trailing zeros in the binary representation of $n$.
Also, let $\operatorname{ntr}(n)$ be A086784, number of non-trailing zeros in the binary ...
- 2,790
3
votes
1
answer
127
views
"Geodesic coherent" partition of a graph
Let $G=(V,E)$ be a finite undirected graph which we equip with its usual graph geodesic distance $d_G$ making $(G,d_G)$ into a metric space; let $1<\#V<\infty$. For a given $1<N< \#V$ ...
- 4,961
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votes
1
answer
87
views
Partition of $(2^{n+1}+1)2^{2^{n-1}+n-1}-1$ into parts with binary weight equals $2^{n-1}+n$
Let $\operatorname{wt}(n)$ be A000120, i.e., number of $1$'s in binary expansion of $n$ (or the binary weight of $n$).
Let $a(n,m)$ be the sequence of numbers $k$ such that $\operatorname{wt}(k)=m$. ...
- 2,790
1
vote
0
answers
149
views
Bounds for the number of self-conjugate partitions [closed]
is there any upper bound for the number of self-conjugate partitions, or equivalently the number of partitions with distinct odd parts?
P.S.: I didn't get why my question has been moved to Math Stack ...
- 531
4
votes
0
answers
187
views
Non-crossing and crossing bijection in higher genus
This is a follow-up question of my SO post I'll briefly mention it here.
So given a $n$ cycle say $(1,2,\ldots,n)$, what are the monotonic 2 -tuples, of the form $(a,b)(c,d)$, monotonicity in on the ...
- 675
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votes
2
answers
498
views
Generating function for counting partitions with corners
A corner of an integer partition is a location at where a box can be added to its Ferrers diagram to give a new partition.
E.g. the partition $\{1,1,1\}$ has two corners, and $\{1,2\}$ has three ...
- 123
0
votes
0
answers
65
views
What will be the smallest value of $k$ such that $P(k)=m$?
Let us suppose that we have a group of order $p^k$, where $p$ is prime.In General,there is one group group of order $p^k$ for each set of positive integers whose sum is $k$(such a set is called ...
- 113
2
votes
0
answers
126
views
Counting integer partitions below some Young diagram
Question: Given positive, coprime integers $m<n$, consider the Young diagram $Y$ formed by the lattice points in the Cartesian plane lying below the line from $(0,0)$ to $(m,n)$ and within the ...
- 705
0
votes
1
answer
119
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Seeking a bijective proof enumerating two partition sets: Part II
An integer partition is a sequence $\lambda=(\lambda_1\geq\lambda_2\geq\dotsb\geq\lambda_k)$ of positive integers, for some $k\geq1$. Consider the following two sets of partitions of $n$. Fix a ...
- 41k
2
votes
1
answer
173
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Seeking a bijective proof enumerating two partition sets: Part I
An integer partition is a sequence $\lambda=(\lambda_1\geq\lambda_2\geq\dotsb\geq\lambda_k)$ of positive integers, for some $k\geq1$. Consider the following two sets of partitions of $n$. Fix a ...
- 41k