Given positive integers $n$ and $d$, let $S$ indicate the list of all $d$-tuples of non-negative integers $(c_1,\ldots,c_d)$ such that $c_1+\cdots+c_d=n$. Let $v_i$ be the value of the multinomial coefficient corresponding to $i$'th tuple in $S$, ie

$$v_i=\frac{n!}{c_1!\cdots c_d!}$$

What can we say about the sum of smallest coefficients, ie, the value of the following?

$$s(B)=\sum_{v_i < B} v_i$$

Motivation: upper bounds on multinomial tails would allow to give non-asymptotic error bounds for various learning algorithms

**Update: 09/03**
Here are all the relevant theoretical results I found so far. Let $B=\frac{n!}{c_1!\cdots c_d}$, $C=\max_i v_i$, $k=\min_i c_i$. Then for even n and $d=2$ the following are known to hold

$$s(B)<\frac{B}{C} 2^n$$ Proof under Lemma 3.8.2 of Lovasz et al "Discrete Mathematics" (2003)

$$s(B)\le 2^n \exp(-\frac{(n/d-k)^2}{n-k})$$ Proof under Theorem 5.3.2 of Lovasz et al "Discrete Mathematics" (2003)

$$s(B)\le 2B(\frac{n-(k-1)}{n-(2k-1)}-1)$$ Michael Lugo gives outline of proof in another MO post

$$s(B)<2(\exp(n \log n - \sum_i c_i \log c_i)-B)$$ Proof under Lemma 16.19 of Flum et al "Parameterized Complexity Theory" (2006)

To be practically useful for my application, these bounds need to be tight for tails, ie, for sums that are less than $d^n/10$. Here's a plot of logarithm of bound/exact ratio for such sums. X-axis is monotonically related to B.

You can see that Michael Lugo's bound is by far the most accurate in that range.

Out of curiosity, I "plugged in" bounds above for sums of higher dimensional coefficients.

Mathematica notebook.