// library begin: "lib/template.hpp" #ifdef TEMPLATE #else #define TEMPLATE # pragma GCC optimize("O3") #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include using namespace std; namespace ryoku{} using uint=unsigned; using ll=long long; using ull=unsigned long long; using ld=long double; using pii=pair; using pll=pair; using i128=__int128; using u128=unsigned __int128; templateusing vc=vector; templateusing vvc=vc>; templateusing vvvc=vvc>; templateusing smpq=priority_queue,greater>; templateusing bipq=priority_queue; #define rep(i,n) for(ll i=0;i<(ll)(n);i++) #define REP(i,j,n) for(ll i=(j);i<(ll)(n);i++) #define DREP(i,n,m) for(ll i=(n);i>=(m);i--) #define drep(i,n) for(ll i=((n)-1);i>=0;i--) #define rall(x) x.rbegin(),x.rend() #define mp(...) make_pair(__VA_ARGS__) #define pb push_back #define fi first #define se second #define is insert #define bg begin() #define ed end() #define all(x) x.begin(),x.end() void scan(int&a) { cin >> a; } void scan(ll&a) { cin >> a; } void scan(string&a) { cin >> a; } void scan(char&a) { cin >> a; } void scan(uint&a) { cin >> a; } void scan(ull&a) { cin >> a; } void scan(bool&a) { cin >> a; } void scan(ld&a){ cin>> a;} template void scan(vector&a) { for(auto&x:a) scan(x); } void read() {} template void read(Head&head, Tail&... tail) { scan(head); read(tail...); } #define INT(...) int __VA_ARGS__; read(__VA_ARGS__); #define LL(...) ll __VA_ARGS__; read(__VA_ARGS__); #define ULL(...) ull __VA_ARGS__; read(__VA_ARGS__); #define STR(...) string __VA_ARGS__; read(__VA_ARGS__); #define VC(type, name, ...) vector name(__VA_ARGS__); read(name); #define VVC(type, name, size, ...) vector> name(size, vector(__VA_ARGS__)); read(name); templatevoid print(T a) { cout << a; } template void print(vectora) { for(int i=0;i<(int)a.size();i++){if(i)cout<<" ";print(a[i]);}} void PRT() { cout < void PRT(T a) { print(a); cout < void PRT(Head head, Tail ... tail) { print(head); cout << " "; PRT(tail...); return; } template bool chmin(T &x, F y){ if(x>y){ x=y; return true; } return false; } template bool chmax(T &x, F y){ if(x T floor(T a,T b){ return a/b-(a%b&&((a<0)!=(b<0))); } template T ceil(T a,T b){ return a/b+(a%b&&((a<0)==(b<0))); } template T bmod(T x,T y){ return x-y*floor(x,y); } template pairdivmod(T x,T y){ T q=floor(x,y); return{q,x-q*y}; } void YesNo(bool b){ cout<<(b?"Yes":"No")<stovi(const string&s,const string&S){ vcv(s.size()); rep(i,s.size()){ auto t=S.find(s[i]); assert(t!=string::npos); v[i]=t; } return v; } template T isqrt(T x){ T F=sqrtl(x); while((F+1)*(F+1)<=x)F++; while(F*F>x)F--; return F; } template T tri(T x){return x*(x-1)/2;} //[l,r) templateT tri(T l,T r){return (r-l)*(l+r-1)/2;} //n を先頭に持ってくる template vcrot(vcv,int n){ rotate(v.begin(),v.begin()+n,v.end()); return v; } template auto rle(const Container&cv){ using T=typename Container::value_type; vc>res; for(const auto&x:cv){ if(res.empty()||res.back().fi!=x)res.pb({x,1}); else res.back().se++; } return res; } template vciota(int n){ vcv(n);rep(i,n)v[i]=i; return v; } template vcargsort(const vc&a){ auto idx=iota(a.size()); sort(all(idx),[&](int i,int j){ return (minfirst?make_pair(a[i],i)make_pair(a[j],j)); }); return idx; } template vvctrans(const vvc&a){ assert(a.size()&&a[0].size()); rep(i,a.size())assert(a[i].size()==a[0].size()); vvcb(a[0].size(),vc(a.size())); rep(i,a.size())rep(j,a[0].size())b[j][i]=a[i][j]; return b; } vctrans(const vc&a){ assert(a.size()&&a[0].size()); rep(i,a.size())assert(a[i].size()==a[0].size()); vcb(a[0].size(),string(a.size(),0)); rep(i,a.size())rep(j,a[0].size())b[j][i]=a[i][j]; return b; } template int popcount(T n){ return __builtin_popcountll(n); } template L sum(const vc&a){ return accumulate(all(a),L(0)); } template struct subset_view{ T s; struct iterator{ T s,x; bool done; T operator*()const{return x;} iterator&operator++(){ if(x==0)done=true; else x=(x-1)&s; return*this; } bool operator!=(const iterator&r)const{return done!=r.done;} }; iterator begin()const{return{s,s,false};} iterator end()const{return{s,0,true};} }; template subset_viewsubset(T s){ return{s}; } template T max(vc&a){ return *max_element(all(a)); } template T min(vc&a){ return *min_element(all(a)); } template vc presum(vc &a){ vc ret(a.size()+1); rep(i,a.size())ret[i+1]=ret[i]+a[i]; return ret; } template vc &operator+=(vc &a,F b){ for (auto&v:a)v += b; return a; } template vc &operator-=(vc&a,F b){ for (auto&v:a)v-=b; return a; } template vc &operator*=(vc&a,F b){ for (auto&v:a)v*=b; return a; } template constexpr T pow(T a,T b){ T res=1; while(b){ if(b&1)res*=a; a*=a; b/=2; } return res; } constexpr ll ten(ll a){ return pow(10,a); } templateconstexpr T inf=numeric_limits::max()/2-1; template<> constexpr pll inf ={inf,inf}; template constexpr T neg_inf=-inf; template<> constexpr pll neg_inf ={-inf,-inf}; template int tbit(T x){ using U=make_unsigned_t; U y=(U)x; return y?(int)bit_width(y)-1:-1; } template int lbit(T x){ using U=make_unsigned_t; U y=(U)x; return y?(int)countr_zero(y):-1; } template int tbit(T x,int p){ using U=make_unsigned_t; constexpr int W=numeric_limits::digits; U y=(U)x; if(p<0)return -1; if(p>=W-1)return tbit(y); return tbit(y&((U(1)<<(p+1))-1)); } template int lbit(T x,int p){ using U=make_unsigned_t; constexpr int W=numeric_limits::digits; U y=(U)x; if(p<0)return lbit(y); if(p>=W)return -1; return lbit(y&(~U(0)<>(istream&is,i128&x){ string s;is>>s; x=0; int i=0,neg=0; if(s[0]=='-')neg=1,i=1; for(;i<(int)s.size();i++)x=x*10+s[i]-'0'; if(neg)x=-x; return is; } ostream& operator<<(ostream&os,i128 x){ if(x==0)return os<<0; if(x<0)os<<"-"; u128 y=x<0?-(u128)x:(u128)x; string s; while(y)s.pb('0'+y%10),y/=10; reverse(all(s)); return os<sync_with_stdio(0); #ifdef LOCAL cout< struct Sum{ using value_type=T; static constexpr bool commute=true; static value_type op(value_type a,value_type b){return a+b;} static value_type id(){return T(0);} static value_type e(){return id();} }; template struct Prod{ using value_type=T; static constexpr bool commute=false; static value_type op(value_type a,value_type b){return a*b;} static value_type id(){return T(1);} static value_type e(){return id();} }; } // library end: "lib/ds/monoid/sum-prod.hpp" // library begin: "lib/ds/ordered/adjacent-manager.hpp" namespace ryoku{ template struct adjacent_manager{ setsts; using F=function; F add,del; adjacent_manager(F add,F del):add(add),del(del){} bool insert(const key&x){ auto r=sts.lower_bound(x); if(r!=sts.end()&&*r==x)return false; auto l=r==sts.begin()?sts.end():prev(r); if(l!=sts.end()&&r!=sts.end())del(*l,*r); auto it=sts.insert(r,x); if(l!=sts.end())add(*l,*it); if(r!=sts.end())add(*it,*r); return true; } bool erase(const key&x){ auto it=sts.find(x); if(it==sts.end())return false; auto l=it==sts.begin()?sts.end():prev(it); auto r=next(it); if(l!=sts.end())del(*l,*it); if(r!=sts.end())del(*it,*r); sts.erase(it); if(l!=sts.end()&&r!=sts.end())add(*l,*r); return true; } }; } // library end: "lib/ds/ordered/adjacent-manager.hpp" // library begin: "lib/math/modular/binom.hpp" // library include: "lib/template.hpp" namespace ryoku{ template struct binom_has_get_mod:false_type{}; template struct binom_has_get_mod>:true_type{}; template struct binom{ private: static vector&fact_table(){static vectorv={1};return v;} static vector&invfact_table(){static vectorv={1};return v;} static vector&invs_table(){static vectorv={0};return v;} static int&built_mod(){static int mod=-1;return mod;} public: static void build(int n){ auto&_fact=fact_table(); auto&_invfact=invfact_table(); auto&_invs=invs_table(); if constexpr(binom_has_get_mod::value){ auto mod=mint::get_mod(); if(built_mod()!=mod){ _fact={1}; _invfact={1}; _invs={0}; built_mod()=mod; } } if(n<(int)_fact.size())return; int old=_fact.size(); _fact.resize(n+1); _invfact.resize(n+1); _invs.resize(n+1); if constexpr(binom_has_get_mod::value){ auto mod=mint::get_mod(); for(int i=old;i<=n;i++){ _fact[i]=_fact[i-1]*i; if(i==1)_invs[i]=1; else _invs[i]=-_invs[mod%i]*(mod/i); _invfact[i]=_invfact[i-1]*_invs[i]; } }else{ for(int i=old;i<=n;i++){ _fact[i]=_fact[i-1]*i; _invs[i]=mint(1)/i; _invfact[i]=_invfact[i-1]*_invs[i]; } } } static mint fact(int i){ assert(i>=0); build(i); return fact_table()[i]; } static mint invfact(int i){ assert(i>=0); build(i); return invfact_table()[i]; } static mint inv(int i){ assert(i>0); build(i); return invs_table()[i]; } static mint C(int a,int b){//aCb if(b==0)return 1; if(a<0||b<0||a-b<0)return mint(0); build(a); auto&_fact=fact_table(); auto&_invfact=invfact_table(); return _fact[a]*_invfact[b]*_invfact[a-b]; } static mint iC(int a,int b){//1/aCb if(b==0)return 1; if(a<0||b<0||a-b<0)return mint(0); build(a); auto&_fact=fact_table(); auto&_invfact=invfact_table(); return _fact[b]*_fact[a-b]*_invfact[a]; } static mint P(int a,int b){ if(a; static value_type op(value_type a,value_type b){ return {a.first*b.first,b.first*a.second+b.second}; } static value_type id(){ return {1,0}; } };*/ namespace ryoku{ template struct segtree{ using value_type=typename info::value_type; vcnode; int N; int n; int lg; void update(int i){ assert(0=0); N=N_; n=1; lg=0; while(nrequires is_invocable_r_v segtree(int N_,F f){ assert(N_>=0); N=N_; n=1; lg=0; while(n segtree(const vc&val){ *this=segtree(val.size(),[&](int i){ return val[i]; }); } void push_back(value_type val){ if(n==N){ vcvs(n+1);rep(i,n)vs[i]=node[i+n]; vs[n]=val; (*this)=segtree(N+1,[&](auto i){ return vs[i]; }); }else{ node[N+n]=val; int i=N+n; while(i>>=1)update(i);N++; } } void set(int i,value_type val){ assert(0<=i&&i>=1)update(i); } void apply(int i,value_type val){ assert(0<=i&&i>=1)update(i); } value_type prod(int l,int r){ assert(0<=l&&l<=r&&r<=N); l+=n,r+=n; value_type sml=info::id(),smr=info::id(); while(l int max_right(int L,F f){ assert(0<=L&&L<=N); int l=n+L,w=1; value_type ansL=info::id(); for(;L+w<=N;l>>=1,w<<=1)if(l&1){ if(!f(info::op(ansL,node[l])))break; ansL=info::op(ansL,node[l++]); L+=w; } while(w>1){ l<<=1,w>>=1; if(L+w<=N&&f(info::op(ansL,node[l]))){ ansL=info::op(ansL,node[l++]); L+=w; } } return L; } template int min_left(int R,F f){ assert(0<=R&&R<=N); int r=n+R,w=1; value_type ansR=info::id(); for(;R-w>=0;r>>=1,w<<=1)if(r&1){ if(!f(info::op(node[r-1],ansR)))break; ansR=info::op(node[--r],ansR); R-=w; } while(w>1){ r<<=1,w>>=1; if(R-w>=0&&f(info::op(node[r-1],ansR))){ ansR=info::op(node[r-1],ansR); R-=w; r--; } } return R; } }; } #ifdef LOCAL #include"lib/debug/formatters/ds--segment_tree--segtree.hpp" #endif // library end: "lib/ds/segment_tree/segtree.hpp" // library begin: "lib/math/modular/static-mod-int.hpp" // library include: "lib/template.hpp" namespace ryoku{ template struct static_modint{ static_assert(0=0); static_modint mi; mi.val=v; return mi; } static_modint &operator+=(const static_modint&m){ u32 x=val+m.val-mod; val=x+(mod&-(x>>31)); return *this; } static_modint &operator-=(const static_modint&m){ u32 x=val-m.val; val=x+(mod&-(x>>31)); return *this; } static_modint &operator*=(const static_modint&m){ val=u64(val)*m.val%mod; return *this; } static_modint &operator/=(const static_modint&m){ val=u64(val)*m.inv().val%mod; return *this; } static_modint operator-() const{ return static_modint(mod-val); } static_modint operator+() const { return *this; } friend static_modint operator+(static_modint lhs, const static_modint& rhs){ return lhs+=rhs; } friend static_modint operator-(static_modint lhs, const static_modint& rhs){ return lhs-=rhs; } friend static_modint operator*(static_modint lhs, const static_modint& rhs){ return lhs*=rhs; } friend static_modint operator/(static_modint lhs,const static_modint&rhs){ return lhs/=rhs; } bool operator==(const static_modint&p) const{ return p.val==val; } bool operator!=(const static_modint&p) const{ return p.val!=val; } static_modint pow(int64_t n) const{ static_modint res(1),mul(val); while(n){ if(n%2)res*=mul; mul*=mul; n/=2; } return res; } friend ostream&operator<<(ostream&os,const static_modint&p){ os<>(istream&is,static_modint&p){ int64_t x; is>>x; p=static_modint(x); return is; } static_modint inv()const{ int64_t a=val,b=mod,u=1,v=0,t; #ifdef LOCAL assert(gcd(a,b)==1); #endif while(b>0){ t=a/b; swap(a-=t*b,b); swap(u-=t*v,v); } return static_modint(u); } }; } // library end: "lib/math/modular/static-mod-int.hpp" using namespace ryoku; using mint=static_modint<998244353>; using B=binom; void solve(){ LL(n,q); STR(s); segtree>seg(n);rep(i,n)seg.set(i,1); // x-1 | x で変化するという情報を入れておく //[l,r) でずっと same auto add=[&](int l,int r){ if(r==n)return; mint ans=0; { int N=r-l; if(N%2)ans=0; else ans+=B::C(N,N/2)/(N/2+1); } seg.set(r,ans); }; //[l,r) でずっと same auto erase=[&](int l,int r){ if(r==n)return; seg.set(r,1); }; adjacent_managera(add,erase); a.insert(0);a.insert(n); vcval(n); auto flip=[&](int i){ if(i+10&&val[n-1]==0){ PRT(0); continue; } } int last_pos=*next(a.sts.rbegin()); auto dp=seg.all_prod(); mint ans=0; k=abs(k); rep(_,1){ int w=n-last_pos; dbg(w,k,dp); //(0,0) -> (w,K) への dyck path if((w+k)%2)continue; int x=(w+k)/2,y=(w-k)/2; if(x<0||y<0)continue; if(x==0&&y==0){ ans+=dp; }else if(k>0){ ans+=dp*(B::C(w,y)-(y?B::C(w,y-1):mint(0))); }else{ assert(x==y); ans+=dp*B::C(2*x,x)/(x+1); } } PRT(ans); } } } signed main(){ int t=1; // cin >> t; while(t--)solve(); }