# The first E ABBA in 2019

### The main idea of the topic

Ask how many strings of (n + m) A's and (n + m) B's can be divided into (n + m) subsequences of length 2, in which there are exactly n "AB" and M "BA".

### Analysis 2 (broken line method of catteland number)

There are four situations:

1. M = 0 & & n = 0: the answer is 1.
2. M > 0 & & n = 0: at this time, it degenerates into a pure Cattelan number problem. The answer is \${{2m} \ choose {m} - {{2m} \ choose {m - 1} \$.
3. M = 0 & & n > 0: at this time, it degenerates into a pure cartelan number problem. The answer is \${{2n}\choose{n} - {{2n}\choose{n - 1} \$.
4. M > 0 & & n > 0: This is the change of Cattelan number, corresponding to the change from one line to two upper and lower boundary lines on the line graph. The answer is \${2(n + m)}\choose{n + m} - {2(n + m)}\choose{m - 1} - {2(n + m)}\choose{n - 1} \$.

### The code is as follows

```  1 #include <bits/stdc++.h>
2 using namespace std;
3
4 #define INIT() ios::sync_with_stdio(false);cin.tie(0);cout.tie(0);
5 #define Rep(i,n) for (int i = 0; i < (n); ++i)
6 #define For(i,s,t) for (int i = (s); i <= (t); ++i)
7 #define rFor(i,t,s) for (int i = (t); i >= (s); --i)
8 #define ForLL(i, s, t) for (LL i = LL(s); i <= LL(t); ++i)
9 #define rForLL(i, t, s) for (LL i = LL(t); i >= LL(s); --i)
10 #define foreach(i,c) for (__typeof(c.begin()) i = c.begin(); i != c.end(); ++i)
11 #define rforeach(i,c) for (__typeof(c.rbegin()) i = c.rbegin(); i != c.rend(); ++i)
12
13 #define pr(x) cout << #x << " = " << x << "  "
14 #define prln(x) cout << #x << " = " << x << endl
15
16 #define LOWBIT(x) ((x)&(-x))
17
18 #define ALL(x) x.begin(),x.end()
19 #define INS(x) inserter(x,x.begin())
20 #define UNIQUE(x) x.erase(unique(x.begin(), x.end()), x.end())
21 #define REMOVE(x, c) x.erase(remove(x.begin(), x.end(), c), x.end()); // ?? x ?????? c
22 #define TOLOWER(x) transform(x.begin(), x.end(), x.begin(),::tolower);
23 #define TOUPPER(x) transform(x.begin(), x.end(), x.begin(),::toupper);
24
25 #define ms0(a) memset(a,0,sizeof(a))
26 #define msI(a) memset(a,inf,sizeof(a))
27 #define msM(a) memset(a,-1,sizeof(a))
28
29 #define MP make_pair
30 #define PB push_back
31 #define ft first
32 #define sd second
33
34 template<typename T1, typename T2>
35 istream &operator>>(istream &in, pair<T1, T2> &p) {
36     in >> p.first >> p.second;
37     return in;
38 }
39
40 template<typename T>
41 istream &operator>>(istream &in, vector<T> &v) {
42     for (auto &x: v)
43         in >> x;
44     return in;
45 }
46
47 template<typename T>
48 ostream &operator<<(ostream &out, vector<T> &v) {
49     Rep(i, v.size()) out << v[i] << " \n"[i == v.size()];
50     return out;
51 }
52
53 template<typename T1, typename T2>
54 ostream &operator<<(ostream &out, const std::pair<T1, T2> &p) {
55     out << "[" << p.first << ", " << p.second << "]" << "\n";
56     return out;
57 }
58
59 inline int gc(){
60     static const int BUF = 1e7;
61     static char buf[BUF], *bg = buf + BUF, *ed = bg;
62
63     if(bg == ed) fread(bg = buf, 1, BUF, stdin);
64     return *bg++;
65 }
66
67 inline int ri(){
68     int x = 0, f = 1, c = gc();
69     for(; c<48||c>57; f = c=='-'?-1:f, c=gc());
70     for(; c>47&&c<58; x = x*10 + c - 48, c=gc());
71     return x*f;
72 }
73
74 template<class T>
75 inline string toString(T x) {
76     ostringstream sout;
77     sout << x;
78     return sout.str();
79 }
80
81 inline int toInt(string s) {
82     int v;
83     istringstream sin(s);
84     sin >> v;
85     return v;
86 }
87
88 //min <= aim <= max
89 template<typename T>
90 inline bool BETWEEN(const T aim, const T min, const T max) {
91     return min <= aim && aim <= max;
92 }
93
94 typedef long long LL;
95 typedef unsigned long long uLL;
96 typedef pair< double, double > PDD;
97 typedef pair< int, int > PII;
98 typedef pair< int, PII > PIPII;
99 typedef pair< string, int > PSI;
100 typedef pair< int, PSI > PIPSI;
101 typedef set< int > SI;
102 typedef set< PII > SPII;
103 typedef vector< int > VI;
104 typedef vector< double > VD;
105 typedef vector< VI > VVI;
106 typedef vector< SI > VSI;
107 typedef vector< PII > VPII;
108 typedef map< int, int > MII;
109 typedef map< int, string > MIS;
110 typedef map< int, PII > MIPII;
111 typedef map< PII, int > MPIII;
112 typedef map< string, int > MSI;
113 typedef map< string, string > MSS;
114 typedef map< PII, string > MPIIS;
115 typedef map< PII, PII > MPIIPII;
116 typedef multimap< int, int > MMII;
117 typedef multimap< string, int > MMSI;
118 //typedef unordered_map< int, int > uMII;
119 typedef pair< LL, LL > PLL;
120 typedef vector< LL > VL;
121 typedef vector< VL > VVL;
122 typedef priority_queue< int > PQIMax;
123 typedef priority_queue< int, VI, greater< int > > PQIMin;
124 const double EPS = 1e-8;
125 const LL inf = 0x7fffffff;
126 const LL infLL = 0x7fffffffffffffffLL;
127 const LL mod = 1e9 + 7;
128 const int maxN = 1e5 + 7;
129 const LL ONE = 1;
130 const LL evenBits = 0xaaaaaaaaaaaaaaaa;
131 const LL oddBits = 0x5555555555555555;
132
133 LL fac[maxN];
134 void init_fact() {
135     fac[0] = 1;
136     For(i, 1, maxN - 1) fac[i] = (i * fac[i - 1]) % mod;
137 }
138
139 //ax + by = gcd(a, b) = d
140 // Extended Euclid algorithm
141 inline void ex_gcd(LL a, LL b, LL &x, LL &y, LL &d){
142     if (!b) {d = a, x = 1, y = 0;}
143     else{
144         ex_gcd(b, a % b, y, x, d);
145         y -= x * (a / b);
146     }
147 }
148
149 // seek a about p If it does not exist, return-1
150 // a And p Reciprocal, inverse element exists
151 inline LL inv_mod(LL a, LL p = mod){
152     LL d, x, y;
153     ex_gcd(a, p, x, y, d);
154     return d == 1 ? (x % p + p) % p : -1;
155 }
156
157 inline LL comb_mod(LL m, LL n) {
158     LL ret;
159
160     if(m > n) swap(m, n);
161
162     ret = (fac[n] * inv_mod(fac[m], mod)) % mod;
163     ret = (ret * inv_mod(fac[n - m], mod)) % mod;
164
165     return ret;
166 }
167
168 LL n, m, ans;
169
170 int main(){
171     //freopen("MyOutput.txt","w",stdout);
172     //freopen("input.txt","r",stdin);
173     //INIT();
174     init_fact();
175     while(~scanf("%lld%lld", &n, &m)) {
176         ans = comb_mod(n + m, 2 * (n + m));
177         if(n) ans -= comb_mod(n - 1, 2 * (n + m));
178         if(m) ans -= comb_mod(m - 1, 2 * (n + m));
179         ans = (ans + 2 * mod) % mod;
180         printf("%lld\n", ans);
181     }
182     return 0;
183 }```

Tags: Swift iOS

Posted on Thu, 17 Oct 2019 11:53:58 -0700 by mingmangat