Numbers Expressed as Sums of Binomial Coefficients

Definition
Let $n \in \Z_{> 0}$ be a (strictly) positive integer.

Then for all $k \in \Z_{> 0}$, it is possible to express $k$ uniquely in the form:

such that $0 \le k_1 < k_2 < \cdots < k_n$.

Existence of Representation
Proof by induction:

For all $k \in \Z_{\ge 0}$, let $\map P k$ be the proposition that it is possible to express $k$ in the form:


 * $\displaystyle k = \sum_{j \mathop = 1}^n \dbinom {k_j} j$

such that $0 \le k_1 < k_2 < \cdots < k_n$.

Basis for the Induction
$\map P 0$ is true, as $\displaystyle 0 = \sum_{j \mathop = 1}^n \dbinom {j - 1} j$.

This is our basis for the induction.

Induction Hypothesis
Now we need to show that, if $\map P m$ is true, where $r \ge 2$, then it logically follows that $\map P {m + 1}$ is true.

So this is our induction hypothesis:

$m$ can be expressed in the form:
 * $\displaystyle m = \sum_{j \mathop = 1}^n \dbinom {m_j} j$

where $0 \le m_1 < m_2 < \cdots < m_n$.

Then we need to show:

$m + 1$ can be expressed in the form:
 * $\displaystyle m = \sum_{j \mathop = 1}^n \dbinom {k_j} j$

where $0 \le k_1 < k_2 < \cdots < k_n$.

Induction Step
This is our induction step:

Suppose the first $c$ $m_j$ are consecutive, that is:


 * $\forall \, j \in \N: 1 \le j < c: m_{j + 1} - m_j = 1$ and $m_{c + 1} - m_c > 1$

Then:

Since $0 \le m_1 < m_2 < \cdots < m_n$, we must have $m_j \ge j - 1$ for each $j \le n$.

Hence $0 \le 0 < 1 < \cdots < c - 2 < m_c - 1 < m_{c + 1} < \cdots < m_n$.

Thus the expression above satisfy the conditions.

So $\map P m \implies \map P {m + 1}$ and the result follows by the Principle of Mathematical Induction.

Therefore:
 * $\displaystyle \forall n \in \Z_{\ge 0}: \sum_{k \mathop = 0}^n \binom {r + k} k = \binom {r + n + 1} n$

Uniqueness of Representation
Suppose $k$ can be expressed in the form:


 * $\displaystyle k = \sum_{j \mathop = 1}^n \dbinom {k_j} j = \sum_{j \mathop = 1}^n \dbinom {m_j} j$

where $0 \le k_1 < k_2 < \cdots < k_n$ and $0 \le m_1 < m_2 < \cdots < m_n$.

$k_i$ and $m_i$ are not all equal.

Let $c$ be the largest integer such that $k_c \ne m_c$.

assume $k_c < m_c$.

Then:

which is a contradiction.

Thus the representation is unique.

Also see

 * Definition:Combinatorial Number System