I can show that $\sum\limits_{k=1}^m{P_k(n)} = P_m(n+m)$ with induction, but I can't figure out how to create an argument based on Ferrers diagrams which proves this equation.
Any hints in the right direction would be appreciated.
2025-01-13 05:57:00.1736747820
Use a Ferrers diagram to prove that $\sum\limits_{k=1}^m{P_k(n)} = P_m(n+m)$
224 Views Asked by Jeffrey https://math.techqa.club/user/jeffrey/detail At
1
There are 1 best solutions below
Related Questions in COMBINATORICS
- How many different games are there in bridge?
- Discrete mathematics, sets, increasing functions
- Number of necklaces of 16 beads with 8 red beads, 4 green beads and 4 yellow beads
- Logic & Reasoning Question
- Delannoy Paths and Pell Sequence Relation
- Combinatorics Problem - Clients using two seperate services
- There are few boxes labelled with $1,2,4,5,7,8,9$ respectively. How many ways to choose $5$ boxes and arranges the boxes in a row.
- Confused by book's given solution to basic combinatorial problem
- How many ways to write a number $n$ as the product of natural numbers $\geq 2$?
- Confused about how to solve basic combinatorial problem
Related Questions in INTEGER-PARTITIONS
- How many ways to write a number $n$ as the product of natural numbers $\geq 2$?
- Is there a formula for the number of equipartitions of $[n]$ into $k$ parts of size $s = n/k$?
- On partition of integers
- On partitions of integers
- How to prove $\prod_{\lambda\vdash n}\prod_im_i(\lambda)!=\prod_{\lambda\vdash n}1^{m_1(\lambda)}2^{m_2(\lambda)}\cdots$
- Formula for how many combinations of powers of 2 sum to $2^n$
- Lexicographic order is a partial order on the the set of all partitions of the positive integer n.
- Number of pairwise non-isomorphic spanning trees of the wheel $W_n$, with restrictions
- How to make a canonical coin system so that greedy solution is the only optimal solution for change-making problem
- Calculate the area under $f(x) = \sqrt x$ on $[0,4]$ by computing the lower Riemann sum for $f$ with the given partition
Related Questions in YOUNG-TABLEAUX
- 321-avoiding permutations and RSK
- Standard representation of $S_5$
- Product of standard and sign representation of $S_5$
- Young diagram for $S_5$
- Confusing partitions of $S_5$ in two different sources
- Representation from Young tabloids
- Young tableaux to Specht polynomial to Irreducible representation for $(1,3,5) \in S_5$
- Young tableaux of partition $3+1+1$ for the conjugacy classes of $S_5$
- Why is the Young symmetrizer non-zero?
- Use a Ferrers diagram to prove that $\sum\limits_{k=1}^m{P_k(n)} = P_m(n+m)$
Trending Questions
- Induction on the number of equations
- How to convince a math teacher of this simple and obvious fact?
- Refuting the Anti-Cantor Cranks
- Find $E[XY|Y+Z=1 ]$
- Determine the adjoint of $\tilde Q(x)$ for $\tilde Q(x)u:=(Qu)(x)$ where $Q:U→L^2(Ω,ℝ^d$ is a Hilbert-Schmidt operator and $U$ is a Hilbert space
- Why does this innovative method of subtraction from a third grader always work?
- What are the Implications of having VΩ as a model for a theory?
- How do we know that the number $1$ is not equal to the number $-1$?
- Defining a Galois Field based on primitive element versus polynomial?
- Is computer science a branch of mathematics?
- Can't find the relationship between two columns of numbers. Please Help
- Is there a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not?
- Identification of a quadrilateral as a trapezoid, rectangle, or square
- A community project: prove (or disprove) that $\sum_{n\geq 1}\frac{\sin(2^n)}{n}$ is convergent
- Alternative way of expressing a quantied statement with "Some"
Popular # Hahtags
real-analysis
calculus
linear-algebra
probability
abstract-algebra
integration
sequences-and-series
combinatorics
general-topology
matrices
functional-analysis
complex-analysis
geometry
group-theory
algebra-precalculus
probability-theory
ordinary-differential-equations
limits
analysis
number-theory
measure-theory
elementary-number-theory
statistics
multivariable-calculus
functions
derivatives
discrete-mathematics
differential-geometry
inequality
trigonometry
Popular Questions
- How many squares actually ARE in this picture? Is this a trick question with no right answer?
- What is the difference between independent and mutually exclusive events?
- Visually stunning math concepts which are easy to explain
- taylor series of $\ln(1+x)$?
- Determine if vectors are linearly independent
- What does it mean to have a determinant equal to zero?
- How to find mean and median from histogram
- Difference between "≈", "≃", and "≅"
- Easy way of memorizing values of sine, cosine, and tangent
- How to calculate the intersection of two planes?
- What does "∈" mean?
- If you roll a fair six sided die twice, what's the probability that you get the same number both times?
- Probability of getting exactly 2 heads in 3 coins tossed with order not important?
- Fourier transform for dummies
- Limit of $(1+ x/n)^n$ when $n$ tends to infinity
HINT: Start with a Ferrers diagram for a partition of $n+m$ into $m$ parts and remove the leftmost column. What remains is a Ferrers diagram for what? Then show that the correspondence is reversible.