Problem:
Score : 800 points

Problem Statement

Given is a string S consisting of 0 and 1. Find the number of strings, modulo 998244353, that can result from applying the following operation on S between 0 and K times (inclusive):

Choose a pair of integers i, j (1\leq i < j\leq |S|) such that the i-th and j-th characters of S are 0 and 1, respectively. Remove the j-th character from S and insert it to the immediate left of the i-th character.

Constraints

1 \leq |S| \leq 300

0 \leq K \leq 10^9

S consists of 0 and 1.

Input

Input is given from Standard Input in the following format:

S K

Output

Find the number of strings, modulo 998244353, that can result from applying the operation on S between 0 and K times (inclusive).

Sample Input 1

0101 1

Sample Output 1

4

Four strings, 0101, 0110, 1001, and 1010, can result.

Sample Input 2

01100110 2

Sample Output 2

14

Sample Input 3

1101010010101101110111100011011111011000111101110101010010101010101 20

Sample Output 3

113434815

Solution in Common Lisp (SBCL 2.0.3) (judge verdict: Accepted)

(in-package :cl-user)
(eval-when (:compile-toplevel :load-toplevel :execute)
  (sb-int:defconstant-eqx opt
    #+swank '(optimize (speed 3) (safety 2))
    #-swank '(optimize (speed 3) (safety 0) (debug 0))
    #'equal)
  #+swank (ql:quickload '(:cl-debug-print :fiveam) :silent t)
  #-swank (set-dispatch-macro-character
           #\# #\> (lambda (s c p) (declare (ignore c p)) `(values ,(read s nil nil t)))))
#+swank (cl-syntax:use-syntax cl-debug-print:debug-print-syntax)

(defmacro define-int-types (&rest bits)
  `(progn
     ,@(mapcar (lambda (b) `(deftype ,(intern (format nil "UINT~A" b)) () '(unsigned-byte ,b))) bits)
     ,@(mapcar (lambda (b) `(deftype ,(intern (format nil "INT~A" b)) () '(signed-byte ,b))) bits)))
(define-int-types 2 4 7 8 15 16 31 32 62 63 64)

(defconstant +mod+ 998244353)

(defmacro dbg (&rest forms)
  #+swank (if (= (length forms) 1)
              `(format *error-output* "~A => ~A~%" ',(car forms) ,(car forms))
              `(format *error-output* "~A => ~A~%" ',forms `(,,@forms)))
  #-swank (declare (ignore forms)))

(declaim (inline println))
(defun println (obj &optional (stream *standard-output*))
  (let ((*read-default-float-format* 'double-float))
    (prog1 (princ obj stream) (terpri stream))))

;; BEGIN_INSERTED_CONTENTS
;;;
;;; Arithmetic operations with static modulus
;;;

(defpackage :cp/mod-operations
  (:use :cl)
  (:export #:define-mod-operations))
(in-package :cp/mod-operations)

;; NOTE: Currently MOD* and MOD+ doesn't apply MOD when the number of
;; parameters is one. For simplicity I won't fix it for now.
(defmacro define-mod-operations (divisor)
  (let ((mod* (intern "MOD*"))
        (mod+ (intern "MOD+"))
        (incfmod (intern "INCFMOD"))
        (decfmod (intern "DECFMOD"))
        (mulfmod (intern "MULFMOD")))
    `(progn
       (defun ,mod* (&rest args)
         (reduce (lambda (x y) (mod (* x y) ,divisor)) args))

       (defun ,mod+ (&rest args)
         (reduce (lambda (x y) (mod (+ x y) ,divisor)) args))

       #+sbcl
       (eval-when (:compile-toplevel :load-toplevel :execute)
         (locally (declare (sb-ext:muffle-conditions warning))
           (sb-c:define-source-transform ,mod* (&rest args)
             (if (null args)
                 1
                 (reduce (lambda (x y) `(mod (* ,x ,y) ,',divisor)) args)))
           (sb-c:define-source-transform ,mod+ (&rest args)
             (if (null args)
                 0
                 (reduce (lambda (x y) `(mod (+ ,x ,y) ,',divisor)) args)))))

       (define-modify-macro ,incfmod (delta)
         (lambda (x y) (mod (+ x y) ,divisor)))

       (define-modify-macro ,decfmod (delta)
         (lambda (x y) (mod (- x y) ,divisor)))

       (define-modify-macro ,mulfmod (multiplier)
         (lambda (x y) (mod (* x y) ,divisor))))))

;; BEGIN_USE_PACKAGE
(eval-when (:compile-toplevel :load-toplevel :execute)
  (use-package :cp/mod-operations :cl-user))
(in-package :cl-user)

;;;
;;; Body
;;;

(define-mod-operations +mod+)
(defun main ()
  (declare #.cl-user::opt)
  (let* ((line (read-line))
         (pos (position #\  line))
         (s (map 'simple-bit-vector #'digit-char-p (subseq line 0 pos)))
         (count1 (count 1 s))
         (k (min count1 (read-from-string line t nil :start pos)))
         (seq (make-array (length s) :element-type 'uint31 :fill-pointer 0)))
    (declare (uint16 k count1))
    (let ((prev -1))
      (dotimes (i (length s))
        (when (zerop (aref s i))
          (assert (vector-push (- (- i prev) 1) seq))
          (setq prev i)))
      (when (= 1 (aref s (- (length s) 1)))
        (assert (vector-push (- (length s) prev 1) seq))))
    (let* ((seq (coerce seq '(simple-array uint31 (*))))
           (n (length seq))
           (cumuls (make-array (+ n 1) :element-type 'uint31 :initial-element 0))
           (dp (make-array (list (+ n 1) (+ count1 1) (+ k 1))
                           :element-type 'uint31
                           :initial-element 0)))
      (declare (uint16 n))
      (loop for i from (- n 1) downto 0
            do (setf (aref cumuls i)
                     (+ (aref cumuls (+ i 1)) (aref seq i))))
      (setf (aref dp 0 0 0) 1)
      #>seq
      #>cumuls
      (dotimes (x n)
        (let ((num (aref seq x)))
          (dotimes (y (+ count1 1))
            (dotimes (z (+ k 1))
              (let ((value (aref dp x y z)))
                (unless (zerop value)
                  (let ((sup (aref cumuls (+ x 1))))
                    (loop for dest-y from y to sup
                          do (incfmod (aref dp (+ x 1) dest-y z) value)))
                  (loop for delta from 1 to (min y num (- k z))
                        do (incfmod (aref dp (+ x 1) (- y delta) (+ z delta))
                                    value))))))))
      (let ((res 0))
        (dotimes (z (+ k 1))
          (incfmod res (aref dp n 0 z)))
        (println res)))))

#-swank (main)

;;;
;;; Test and benchmark
;;;

#+swank
(defun get-clipbrd ()
  (with-output-to-string (out)
    #+os-windows (run-program "powershell.exe" '("-Command" "Get-Clipboard") :output out :search t)
    #+os-unix (run-program "xsel" '("-b" "-o") :output out :search t)))

#+swank (defparameter *this-pathname* (uiop:current-lisp-file-pathname))
#+swank (defparameter *dat-pathname* (uiop:merge-pathnames* "test.dat" *this-pathname*))

#+swank
(defun run (&optional thing (out *standard-output*))
  "THING := null | string | symbol | pathname

null: run #'MAIN using the text on clipboard as input.
string: run #'MAIN using the string as input.
symbol: alias of FIVEAM:RUN!.
pathname: run #'MAIN using the text file as input."
  (let* ((*standard-output* (or out (make-string-output-stream)))
         (res (etypecase thing
                (null
                 (with-input-from-string (*standard-input* (delete #\Return (get-clipbrd)))
                   (main)))
                (string
                 (with-input-from-string (*standard-input* (delete #\Return thing))
                   (main)))
                (symbol (5am:run! thing))
                (pathname
                 (with-open-file (*standard-input* thing)
                   (main))))))
    (if out res (get-output-stream-string *standard-output*))))

#+swank
(defun gen-dat ()
  (uiop:with-output-file (out *dat-pathname* :if-exists :supersede)
    (format out "")))

#+swank
(defun bench (&optional (out (make-broadcast-stream)))
  (time (run *dat-pathname* out)))

;; To run: (5am:run! :sample)
#+swank
(it.bese.fiveam:test :sample
  (it.bese.fiveam:is
   (equal "4
"
          (run "0101 1
" nil)))
  (it.bese.fiveam:is
   (equal "14
"
          (run "01100110 2
" nil)))
  (it.bese.fiveam:is
   (equal "113434815
"
          (run "1101010010101101110111100011011111011000111101110101010010101010101 20
" nil))))