T1
The meaning of problems: \ (n-\) variables, \ (0 \ Leq x_i \ Leq C_i \) , seeking \ (\ sum x_i = A \ ) program number. \ (n-\ Leq 32 \) .
Sol：
\ (n \ leq 10 \) when the volume of water repellency is,However generating function is down。
\ (n \ leq 32 \) when, dls: "Obviously Meet in Middle."Then I dropped the
The world would I not generating function
T2
The meaning of problems: seeking \ (0 \) to \ (2n-1 \) arranged \ (P \) number, such that for any \ (I \) , \ (n-2 ^ \ I ^ 2 + Leq P_i 2 ^ \ Leq 4n ^ 2 \) . \ (. 1 \ n-Leq \ Leq 250 \) .
Sol:
Obviously it can be converted to \ (l_i \ leq p_i \ leq r_i \) form.
Consider only \ (r_i \) restrictions, they can follow \ (r_i \) is sorted, then multiplication principle.
Direct inclusion and exclusion \ (l_i \) does not do, but there are some properties of the subject:
\ (l_i, r_i \) is decreasing, and \ (\ {r_0, ..., r_ {n-1} \} \) maximum, \ (\ {L_0, ...,. 1-n-menthoxypropane {} \} \ ) and \ (\ {r_n, ..., r_ {2n-1} \} \) hybrid.
Conversion look into the following sequence: the first half \ (a, b \) mixing, only the latter half of \ (C \) , and \ (a, c \) paired off between each pair can only be selected from a each \ (b \) Required. The total weight is \ (\ Prod (val_i-rk_i) \ Times (-1) ^ {| A |} \) .
Consider the outer enumeration \ (A \) number, in the inner dp, \ (F_ {I, J} \) represents the former \ (I \) substituents selected \ (J \) a \ (A \) of weight product, so every time no matter what the elements can be calculated rankings.
Finally understood the problems of the qwq
dlstxdy！
T3 Big # 575
Meaning of the questions: Given the size of the relationship between adjacent elements, find the number of permutations. \ (n-\ Leq. 5 ^ 10 \) .
Sol:
DLS: " \ (10 ^ 5 \) is rubbish Law Act tower, consider \ (n ^ 2 \) just fine."
Found = greater than the unrestricted - less than.
To turn into a number greater than or less than the limitation, original sequence into a plurality of segments, each segment is increasing sequence, the answer is clearly \ (\ {n-FRAC!} {\ Prod P_i!} \) ,Just like dp。
Each transfer a whole paragraph, the entire paragraph is greater than the level of a full-push.
T4
The meaning of problems: \ (n-\ Times m \) matrix, both initial \ (0 \) , per row may be selected from a prefix \ (1 + \) , seeking essentially different matrices. \ (n-, m \ Leq. 6 10 ^ \) .
Sol：
dls: "In addition to the rows and columns formed just two prefixes inverted L in both cases, the other will not be repeated."
Proof? naidesu
Wherein the hard spot may be a certain kind of illegal, there are enumerated \ (I \) row \ (I \) column is not valid, receiving the coefficient repellency probably a long way:
\[(-1)^i C(n, i) C(m, i) i!(n+1)^{m-i}(m+1)^{n-i}\]
\ (i! \) is the meaning of \ (i \) rows, each row to choose a mate.
Difficulties are not inclusion-exclusion, and in thatGuess Conclusion。
T5 CTS2019 random cube
Sol：
Conclusion. 1: \ (n-\) number, each number is \ ([0,1] \) random real number between, is equivalent to a \ ([1, n] \ ) is arranged.
Conclusion 2: \ (n-\) tree points, each point corresponding to an arrangement of elements, such as small tree root is formed stack, program number \ (\ frac {\ prod_i ^ n size_i} {n!} \) .
First consider the two-dimensional, three-dimensional to expand later.
Obviously \ (k \) maximum values are not in the same row or column, consider \ (k \) th smallest, its row and column must be less than it. For the second smallest value, there are two rows and two columns to be less than it. So it revealed that a similar tree structure, using the above embodiment the number of conclusions can be calculated.
Extended to three dimensions is also very easy, just you need to take one-dimensional.
But this is not "just" cases, there may be more great value.
Binomial inversion can be.
dls: "This problem is CTS in the kind of simple, although there are national team players will not do."
dlstsdy！
T6
Meaning of the questions: \ (the n-\) points, each point even random two sides expect Unicom to ask after how many steps. \ (n-\ Leq 100 \) .
Sol:
SD Province set talked about, although I almost forgotten
Min-Max receiving a desired repellent is also applicable, because the maximum number of times and take a desired operation can not be simply determined and then take the maximum value of each element Max. But it is expected to do a minimum number of times, because its meaning is the earliest appearance of the time element, simply subtract from the total did not choose this probability to the collection can be.
Min-Max desirable example: \ (n-\) a set of elements, each generating a subset \ (S \) probability of \ (P_S \) , and the desired number of steps to ask the repertoire.
To calculate the probability of each set of "any element not chosen this collection", the direct inversion can be.
dls he told aLooksGod is a connected graph counting practice, probably understand a little,Do not remember the formula is too long
dls: This question and Min-Max inversion does not matter, we San Lebammp
T7
Meaning of the questions: length \ (n \) sequence, each time a random interval Black, asked a few steps expected all black. \ (n-\ Leq 100 \) .
Sol:
Min-Max is also the inclusion-exclusion, and now need to find "the probability of a staining dye less than this collection of" For each collection, but not hard to do.
Suppose the hard point sequence number check, then the dye can be a number of intervals must be dp.
Set \ (f_ {i, j, k} \) is a front \ (I \) elements, the last stretch of consecutive not selected length \ (J \) , a total of \ (K \) legitimate interval of tolerance and denounced coefficient . Enumeration of \ (i \) element of the election do not choose, you can transfer respectively.
Core Min-Max inversion is to consider how to calculate the probability of "not chosen the collections".
T8
Meaning of the questions: \ (the n-\ Times m \) grid, some grid unreachable. From Q \ ((1, 0) \) to \ ((n-1, m ) \) and \ ((0,1) \) to \ ((n, m-1 ) \) two disjoint paths the number of programs.
Sol:
For each intersecting path, from the first flip intersection, a correspondence can be from \ ((1, 0) \) to \ ((n, m-1 ) \) and \ ((0,1) \) to \ ((n-1, m ) \) path (such as path must have an intersection).
It can be extended to \ (\ k) case since the end of the group, but all require conditions "must intersect" in. ( LGV Lemma ) (the last will generate a \ (k \) determinant)
T9
The meaning of problems: the (n \ times m \) \ grid filled in \ ([. 1, K] \) (repeat), requires each element is greater than or equal to the right lower side of the element, the number of required program.
Sol:
Young tables can do, but we want to teach dls LGV Lemma.
The number of each contour line pull out, pan left on a grid, it becomes strictly disjoint pieces of the number of paths.
T10 EI count rhomboid
Sol：
SD Province set also talked about ...... but yfz speak.
This discovery is a perspective view of things, there is the nature of Yang table.
(Seemingly) Yang table solution can be used for things that can turn into LGV Lemma?