All Questions
Tagged with co.combinatorics partitions
300 questions
3
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116
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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 ...
- 10.5k
0
votes
0
answers
121
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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
4
votes
0
answers
145
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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/...
1
vote
0
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170
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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 $...
- 15.6k
3
votes
1
answer
985
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Optimal algorithm for a "round robin" doubles tournament?
I have 4N players ( N = 4 or N = 5 suffices) and want to set up three rounds of play. In each round, there will be N games played (four players per game). I want to set up the groupings so that no ...
- 143
2
votes
0
answers
200
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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 $\lambda$ ...
- 111
1
vote
1
answer
308
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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 ...
- 568
0
votes
1
answer
367
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
196
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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 $...
- 43.2k
5
votes
2
answers
269
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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$?
...
- 1,996
1
vote
0
answers
72
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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\...
- 48.8k
1
vote
2
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788
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 ...
- 4,589
4
votes
1
answer
206
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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$ ...
- 4,928
5
votes
0
answers
135
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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) \, =
\,
\...
- 2,137
1
vote
0
answers
100
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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 ...
- 4,928
3
votes
1
answer
92
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$. ...
- 4,928
1
vote
0
answers
158
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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 ...
- 539
4
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0
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206
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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 ...
- 685
12
votes
2
answers
541
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 ...
- 335
3
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144
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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 ...
- 956
0
votes
1
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132
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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 ...
- 43.2k
2
votes
1
answer
181
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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 ...
- 43.2k
2
votes
0
answers
243
views
Pairs vs. two pieces: is the usual proof model-theoretically-optimal?
(For clarity, I'll use $R,S$ for binary relation symbols and $A,B$ for actual binary relations.)
There is an equality between the numbers (up to isomorphism in the appropriate sense) of partitions of ...
- 21.1k
4
votes
1
answer
349
views
The fraction $\frac{g_{\mu}}{f_{\lambda}}$ is an integer
Let $\lambda=(\lambda_1\geq\lambda_2\geq\cdots\geq\lambda_{\ell(\lambda)}>0)$ be an integer partition of $n\in\mathbb{N}$; i.e., $\lambda_1+\cdots+\lambda_{\ell(\lambda)}=n$.
One may now associate $...
- 43.2k
12
votes
1
answer
596
views
Equality of two $q$-series. Proof?
Recall the notation $(z;q)_n=(1-z)(1-zq)(1-zq^2)\cdots(1-zq^{n-1})$.
My earlier MO question did not find enough interest or yield an answer. Perhaps the modulo $2$ part might have thrown people off. ...
- 43.2k
3
votes
1
answer
251
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Congruence modulo 2 for q-series
This quest arose from certain calculations with integer partitions (having distinct parts) and the corresponding values of their Dyson ranks.
I would like to ask:
QUESTION. Is this congruence true ...
- 43.2k
3
votes
1
answer
220
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Agreement between two sets of data on partitions
Let $\lambda=(\lambda_1,\lambda_2,\dotsc,\lambda_{\ell(\lambda)})$ be an integer partition of positive numbers where $\ell(\lambda)$ is the length of the partition. One may associate a Ferrer diagram ...
- 43.2k
1
vote
0
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72
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Factorizable partition polynomials
Let $p(n)$ denote the number of (unrestricted) integer partition of $n$ which has the product generating function
$$\sum_{n\geq0}p(n)\,x^n=\prod_{j\geq1}\frac1{1-x^j}.$$
On the other hand, for the ...
- 43.2k
4
votes
1
answer
263
views
A refinment of Beck's conjecture
Let $\mathcal{O}(n)$ and $\mathcal{D}(n)$ denote the set of all integer partitions of $n$ into odd parts and distinct parts, respectively. Let $o(n)=\#\mathcal{O}(n)$ and $d(n)=\#\mathcal{D}(n)$. ...
- 43.2k
1
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0
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156
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Number-theoretic proof of integrality of a fraction and asymptotics of sum over partitions related to symmetric group
Consider $\;\alpha=(\alpha_1,...,\alpha_n)\in\mathbb{Z}_+^n\;$ such that $\;1\alpha_1+...+n\alpha_n=n.\;$ Let $\varphi$ denote Euler totient-function.
Let $\;T_\alpha\;$ be a set of permutations in $...
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1
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0
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65
views
Partitioning antidirected trees with bounded degree, such that the graph induced by the partition is a constant antidirected tree
Given a partition of the vertices of a graph, we can define an auxiliary graph which conveys information about the edges between sets of the partition. This defines a graph with vertex set equal to ...
- 71
-2
votes
1
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139
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Congruence modulo 4 for a generating function leads to perfect squares? [duplicate]
Consider the number of integer partitions $p(n)$ of $n$ whose generating function is
$$\sum_{n\geq0}p(n)\,x^n=\prod_{k\geq1}\frac1{1-x^k}.$$
Also, the number of partitions into distinct parts $Q(n)$ ...
- 43.2k
8
votes
3
answers
2k
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Bijective proof for a partition identity
I came across the following cute fact about partitions:
\begin{align}
& |\{\lambda \vdash n \text{ with an even number of even parts}\}| \\[8pt]
& {} - |\{ \lambda \vdash n \text{ with an odd ...
- 2,242
1
vote
0
answers
203
views
Generalizing "partition into odd parts=partition into distinct parts"?
The number of partitions into distinct parts is known to agree with the number of partitions with odd parts. For instance, this follows from
$$\prod_{k=1}^{\infty}(1+q^k)=\prod_{n=1}^{\infty}\frac1{1-...
- 43.2k
2
votes
3
answers
599
views
Infinite product of $1-q^{n^2}$
Is there anything known about the following product? Is it a known function or related to a known function?
$$\prod_{n\geqslant1}(1-q^{n^2})$$
- 539
0
votes
0
answers
75
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Objects equinumerous with $3$-ary partitions?
There is a concept of the so-called RP-compositions of an integer discussed by K. Q. Ji and H. S. Wilf in Extreme palindromes. They proved the following result too:
Theorem. The number of RP-...
- 43.2k
3
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0
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229
views
Combinatorial interpretation of a determinant
This is a continuation of Worpitzky-like identities?.
Let $ r_k(x)=\prod_{j=1}^k {(\frac{x+j}{j}})^{\min(j,k-j)}.$
As Sam Hopkins has remarked $r_k(x)$ is the number of plane partitions in a $ \...
- 5,622
5
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1
answer
223
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Coefficients obtained from ratio with partition number generating function
This is a question inspired by T. Amdeberhan's recent question, as well as another previos MO question.
For an integer partition $\lambda$, and $k\in \mathbb{N}\cup\{\infty\}$, let $|\lambda|_k$ ...
- 24.2k
13
votes
2
answers
803
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Two interpretations of a sequence: an opportunity for combinatorics
The sequence that is addressed here is resourced from the most useful site OEIS, listed as A014153, with a generating function
$$\frac1{(1-x)^2}\prod_{k=1}^{\infty}\frac1{1-x^k}.$$
In particular, look ...
- 43.2k
7
votes
0
answers
193
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Factoring a function from a finite set to itself
Let $S$ be a finite set and $f: S \to S$ be a function. Let $k = |f(S)|$ and let $\alpha$ be the partition of $S$ into $f$-fibers, i.e. $\alpha = \{ \alpha_t \}_{t \in f(S)}$ where $\alpha_t = f^{-1}(\...
- 695
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3
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327
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Partitions that are "mutually nested"
Let $\mathcal{P}_1,\dots,\mathcal{P}_m$ be a collection of ordered $n$-partitions of a set $\mathcal S$, which is to say that that $$\mathcal{P}_i = \{P^i_1\cup\dots\cup P^i_n\}$$ for all $i$. ...
- 4,049
1
vote
0
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82
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How to obtain explicit formula for this sum over Young diagram?
Consider the next essence
$$
B_N (r, q) =\sum_{\tau \vdash r} d^2 (\tau ) \prod_{i = 1}^{r} \frac{\Gamma [N + \tau_i - i +1]}{\Gamma [N + \tau_i - i +1+q]}
$$
where $d(\tau)$ is dimension of ...
2
votes
1
answer
137
views
How to re-expand the sum of Schur function?
Consider next sum
\begin{eqnarray}
\label{PF_spindef}
Z = \sum_{r=0}^{N N_f} h^{2r} \ Q(r) .
\end{eqnarray}
and
\begin{equation}
Q(r) \ = \ \sum_{\sigma \vdash r} s_{\sigma}(1^{N_f}) \
s_{\sigma}...
2
votes
0
answers
168
views
New identity for sum over Young diagram of symmetric group?
Consider the next identity
$$
\sum_{\tau \vdash r} d^2 (\tau ) \! \prod_{i = -(r-1)}^{r-1} \! \! \left( N+i \right)^{t_i^r} =\sum_{\tau \vdash r} d^2 (\tau ) \prod_{i = 1}^{r} \Gamma [N + \tau_i - ...
12
votes
3
answers
892
views
Set partitions and permanents
Let $a(n)=$ Number of ordered set partitions of $[n]$ such that the smallest element of each block is odd.
...
- 884
2
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0
answers
59
views
Relations between LR coefficients and cores and quotients of partitions
I have a formula for certain coefficients in terms of Littlewood-Richardson coefficients and $p$-cores and $p$-quotients of partitions ($p$ is a prime). I would like to obtain some positivity ...
- 21
10
votes
1
answer
272
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Plane partitions as irreducible representations
The irreducible representations of the symmetric group algebras $A_n=KS_n$ over a the complex numbers (or any field of characteristic 0) $K$ satisfy the following properties:
The irreducible ...
- 26.5k
3
votes
0
answers
254
views
Enumerating multi-core binary partitions
An integer partition $\lambda$ of $n$ is called a binary partition provided that its parts are powers of $2$ (dyadic). Example: Let $n=3$. The binary partitions are $\lambda=(2,1)$ and $\lambda=(1,1)$ ...
- 43.2k
3
votes
1
answer
372
views
How many ways can $N$ be written as a sum of terms in the form $2^i3^j$?
Given a positive integer $N$, let $f(N)$ be the number of ways $N$ can be decomposed as a sum of terms of the form $2^i3^j$, where each such term appears at most once in the sum. For example, $f(10) = ...
- 1,703
3
votes
0
answers
123
views
$q$-series for the number of rectangles in a square lattice
Given a partition $\lambda\vdash n$ of $n$, look at its Young diagram $Y_{\lambda}$. Let $a(\lambda)$ be the number of squares (of all sizes) in $Y_{\lambda}$. For example, if $n=4$ then $a(4)=4, a(3,...
- 43.2k