Language: C++

#include <algorithm>
#include <iostream>
#include <cstring>
#include <cstdlib>
#include <cstdio>
#include <cmath>

#define Mod 1000000007
#define Maxn 100010
#define R register
#define N ( n << 1 )
#define ll long long

using namespace std;

int n;
ll ans;
bool vis[Maxn];
int a[Maxn] , r[Maxn] , r2[Maxn] , hash[Maxn];
string s;

inline ll check() {
	
	r2[0] = 0;
	for(R int i=N; i>=1; i--)
		if(!vis[i])
			r2[++r2[0]] = i;
	
	for(R int i=1; i<=n; i++)
		if(a[r[i]] != a[r2[i]])
			return 0ll;
	
	return 1ll;
	
}

void dfs(int step , int now) {
	
	if(step > n) {
		ans += check();
		return ;
	}
	
	for(R int i=now+1; i<=N; i++) {
		vis[i] = 1;
		r[step] = i;
		dfs(step+1,i);
		r[step] = 0;
		vis[i] = 0;
	}
	
}

int main() {
	
	scanf("%d",&n);
	
	cin >> s;
	
	for(R int i=0; i<N; i++) {
		R int tmp = s[i] - 'a' + 1;
		a[i+1] = tmp;
		hash[tmp] ++;
	}
	
	for(R int i=1; i<=26; i++)
		if(hash[i]&1)
			return puts("0") , 0;
	
	dfs(1,0);
	
	printf("%lld\n",ans);
	
    return 0;
    
}

Compiler output:
a.cc: In function 'long long int check()':
a.cc:25:19: warning: ISO C++17 does not allow 'register' storage class specifier [-Wregister]
   25 |         for(R int i=N; i>=1; i--)
      |                   ^
a.cc:29:19: warning: ISO C++17 does not allow 'register' storage class specifier [-Wregister]
   29 |         for(R int i=1; i<=n; i++)
      |                   ^
a.cc: In function 'void dfs(int, int)':
a.cc:44:19: warning: ISO C++17 does not allow 'register' storage class specifier [-Wregister]
   44 |         for(R int i=now+1; i<=N; i++) {
      |                   ^
a.cc: In function 'int main()':
a.cc:60:19: warning: ISO C++17 does not allow 'register' storage class specifier [-Wregister]
   60 |         for(R int i=0; i<N; i++) {
      |                   ^
a.cc:61:23: warning: ISO C++17 does not allow 'register' storage class specifier [-Wregister]
   61 |                 R int tmp = s[i] - 'a' + 1;
      |                       ^~~
a.cc:63:17: error: reference to 'hash' is ambiguous
   63 |                 hash[tmp] ++;
      |                 ^~~~
In file included from /usr/include/c++/14/string_view:50,
                 from /usr/include/c++/14/bits/basic_string.h:47,
                 from /usr/include/c++/14/string:54,
                 from /usr/include/c++/14/bits/locale_classes.h:40,
                 from /usr/include/c++/14/bits/ios_base.h:41,
                 from /usr/include/c++/14/ios:44,
                 from /usr/include/c++/14/ostream:40,
                 from /usr/include/c++/14/iostream:41,
                 from a.cc:2:
/usr/include/c++/14/bits/functional_hash.h:59:12: note: candidates are: 'template<class _Tp> struct std::hash'
   59 |     struct hash;
      |            ^~~~
a.cc:19:36: note:                 'int hash [100010]'
   19 | int a[Maxn] , r[Maxn] , r2[Maxn] , hash[Maxn];
      |                                    ^~~~
a.cc:66:19: warning: ISO C++17 does not allow 'register' storage class specifier [-Wregister]
   66 |         for(R int i=1; i<=26; i++)
      |                   ^
a.cc:67:20: error: reference to 'hash' is ambiguous
   67 |                 if(hash[i]&1)
      |                    ^~~~
/usr/include/c++/14/bits/functional_hash.h:59:12: note: candidates are: 'template<class _Tp> struct std::hash'
   59 |     struct hash;
      |            ^~~~
a.cc:19:36: note:                 'int hash [100010]'
   19 | int a[Maxn] , r[Maxn] , r2[Maxn] , hash[Maxn];
      |                                    ^~~~