Stirling numbers of the first kind
You are encouraged to solve this task according to the task description, using any language you may know.
Stirling numbers of the first kind, or Stirling cycle numbers, count permutations according to their number of cycles (counting fixed points as cycles of length one).
They may be defined directly to be the number of permutations of n elements with k disjoint cycles.
Stirling numbers of the first kind express coefficients of polynomial expansions of falling or rising factorials.
Depending on the application, Stirling numbers of the first kind may be "signed" or "unsigned". Signed Stirling numbers of the first kind arise when the polynomial expansion is expressed in terms of falling factorials; unsigned when expressed in terms of rising factorials. The only substantial difference is that, for signed Stirling numbers of the first kind, values of S1(n, k) are negative when n + k is odd.
Stirling numbers of the first kind follow the simple identities:
S1(0, 0) = 1
S1(n, 0) = 0 if n > 0
S1(n, k) = 0 if k > n
S1(n, k) = S1(n - 1, k - 1) + (n - 1) * S1(n - 1, k) # For unsigned
or
S1(n, k) = S1(n - 1, k - 1) - (n - 1) * S1(n - 1, k) # For signed
- Task
- Write a routine (function, procedure, whatever) to find Stirling numbers of the first kind. There are several methods to generate Stirling numbers of the first kind. You are free to choose the most appropriate for your language. If your language has a built-in, or easily, publicly available library implementation, it is acceptable to use that.
- Using the routine, generate and show here, on this page, a table (or triangle) showing the Stirling numbers of the first kind, S1(n, k), up to S1(12, 12). it is optional to show the row / column for n == 0 and k == 0. It is optional to show places where S1(n, k) == 0 (when k > n). You may choose to show signed or unsigned Stirling numbers of the first kind, just make a note of which was chosen.
- If your language supports large integers, find and show here, on this page, the maximum value of S1(n, k) where n == 100.
- See also
- Related Tasks
ALGOL 68
Uses the Algol 68G LONG LONG INT mode which provides large-precision integers. As the default number of digits is insufficient for the task, the maximum nunber of digits is specified by a pragmatic comment. <lang algol68>BEGIN
# show some (unsigned) Stirling numbers of the first kind #
# specify the precision of LONG LONG INT, we need about 160 digits # # for Stirling numbers of the first kind with n, k = 100 # PR precision 160 PR MODE SINT = LONG LONG INT;
# returns a triangular matrix of Stirling numbers up to max n, max n #
# the numbers are signed if signed is TRUE, unsigned otherwise #
PROC make s1 = ( INT max n, BOOL signed )REF[,]SINT:
BEGIN
REF[,]SINT s1 := HEAP[ 0 : max n, 0 : max n ]SINT;
FOR n FROM 0 TO max n DO FOR k FROM 0 TO max n DO s1[ n, k ] := 0 OD OD;
s1[ 0, 0 ] := 1;
FOR n FROM 1 TO max n DO s1[ n, 0 ] := 0 OD;
FOR n FROM 1 TO max n DO
FOR k FROM 1 TO n DO
SINT s1 term = ( ( n - 1 ) * s1[ n - 1, k ] );
s1[ n, k ] := s1[ n - 1, k - 1 ] + IF signed THEN - s1 term ELSE s1 term FI
OD
OD;
s1
END # make s1 # ;
# task requirements: #
# print Stirling numbers up to n, k = 12 #
BEGIN
INT max stirling = 12;
REF[,]SINT s1 = make s1( max stirling, FALSE );
print( ( "Unsigned Stirling numbers of the first kind:", newline ) );
print( ( " k" ) );
FOR k FROM 0 TO max stirling DO print( ( whole( k, -10 ) ) ) OD;
print( ( newline, " n", newline ) );
FOR n FROM 0 TO max stirling DO
print( ( whole( n, -2 ) ) );
FOR k FROM 0 TO n DO
print( ( whole( s1[ n, k ], -10 ) ) )
OD;
print( ( newline ) )
OD
END;
# find the maximum Stirling number with n = 100 #
BEGIN
INT max stirling = 100;
REF[,]SINT s1 = make s1( max stirling, FALSE );
SINT max 100 := 0;
FOR k FROM 0 TO max stirling DO
IF s1[ max stirling, k ] > max 100 THEN max 100 := s1[ max stirling, k ] FI
OD;
print( ( "Maximum Stirling number of the first kind with n = 100:", newline ) );
print( ( whole( max 100, 0 ), newline ) )
END
END</lang>
- Output:
Unsigned Stirling numbers of the first kind: k 0 1 2 3 4 5 6 7 8 9 10 11 12 n 0 1 1 0 1 2 0 1 1 3 0 2 3 1 4 0 6 11 6 1 5 0 24 50 35 10 1 6 0 120 274 225 85 15 1 7 0 720 1764 1624 735 175 21 1 8 0 5040 13068 13132 6769 1960 322 28 1 9 0 40320 109584 118124 67284 22449 4536 546 36 1 10 0 362880 1026576 1172700 723680 269325 63273 9450 870 45 1 11 0 3628800 10628640 12753576 8409500 3416930 902055 157773 18150 1320 55 1 12 0 39916800 120543840 150917976 105258076 45995730 13339535 2637558 357423 32670 1925 66 1 Maximum Stirling number of the first kind with n = 100: 19710908747055261109287881673376044669240511161402863823515728791076863288440277983854056472903481625299174865860036734731122707870406148096000000000000000000
C
<lang c>#include <stdbool.h>
- include <stdio.h>
- include <stdlib.h>
typedef struct stirling_cache_tag {
int max; int* values;
} stirling_cache;
int stirling_number1(stirling_cache* sc, int n, int k) {
if (k == 0)
return n == 0 ? 1 : 0;
if (n > sc->max || k > n)
return 0;
return sc->values[n*(n-1)/2 + k - 1];
}
bool stirling_cache_create(stirling_cache* sc, int max) {
int* values = calloc(max * (max + 1)/2, sizeof(int));
if (values == NULL)
return false;
sc->max = max;
sc->values = values;
for (int n = 1; n <= max; ++n) {
for (int k = 1; k <= n; ++k) {
int s1 = stirling_number1(sc, n - 1, k - 1);
int s2 = stirling_number1(sc, n - 1, k);
values[n*(n-1)/2 + k - 1] = s1 + s2 * (n - 1);
}
}
return true;
}
void stirling_cache_destroy(stirling_cache* sc) {
free(sc->values); sc->values = NULL;
}
void print_stirling_numbers(stirling_cache* sc, int max) {
printf("Unsigned Stirling numbers of the first kind:\nn/k");
for (int k = 0; k <= max; ++k)
printf(k == 0 ? "%2d" : "%10d", k);
printf("\n");
for (int n = 0; n <= max; ++n) {
printf("%2d ", n);
for (int k = 0; k <= n; ++k)
printf(k == 0 ? "%2d" : "%10d", stirling_number1(sc, n, k));
printf("\n");
}
}
int main() {
stirling_cache sc = { 0 };
const int max = 12;
if (!stirling_cache_create(&sc, max)) {
fprintf(stderr, "Out of memory\n");
return 1;
}
print_stirling_numbers(&sc, max);
stirling_cache_destroy(&sc);
return 0;
}</lang>
- Output:
Unsigned Stirling numbers of the first kind: n/k 0 1 2 3 4 5 6 7 8 9 10 11 12 0 1 1 0 1 2 0 1 1 3 0 2 3 1 4 0 6 11 6 1 5 0 24 50 35 10 1 6 0 120 274 225 85 15 1 7 0 720 1764 1624 735 175 21 1 8 0 5040 13068 13132 6769 1960 322 28 1 9 0 40320 109584 118124 67284 22449 4536 546 36 1 10 0 362880 1026576 1172700 723680 269325 63273 9450 870 45 1 11 0 3628800 10628640 12753576 8409500 3416930 902055 157773 18150 1320 55 1 12 0 39916800 120543840 150917976 105258076 45995730 13339535 2637558 357423 32670 1925 66 1
C++
<lang cpp>#include <algorithm>
- include <iomanip>
- include <iostream>
- include <map>
- include <gmpxx.h>
using integer = mpz_class;
class unsigned_stirling1 { public:
integer get(int n, int k);
private:
std::map<std::pair<int, int>, integer> cache_;
};
integer unsigned_stirling1::get(int n, int k) {
if (k == 0)
return n == 0 ? 1 : 0;
if (k > n)
return 0;
auto p = std::make_pair(n, k);
auto i = cache_.find(p);
if (i != cache_.end())
return i->second;
integer s = get(n - 1, k - 1) + (n - 1) * get(n - 1, k);
cache_.emplace(p, s);
return s;
}
void print_stirling_numbers(unsigned_stirling1& s1, int n) {
std::cout << "Unsigned Stirling numbers of the first kind:\nn/k";
for (int j = 0; j <= n; ++j) {
std::cout << std::setw(j == 0 ? 2 : 10) << j;
}
std::cout << '\n';
for (int i = 0; i <= n; ++i) {
std::cout << std::setw(2) << i << ' ';
for (int j = 0; j <= i; ++j)
std::cout << std::setw(j == 0 ? 2 : 10) << s1.get(i, j);
std::cout << '\n';
}
}
int main() {
unsigned_stirling1 s1;
print_stirling_numbers(s1, 12);
std::cout << "Maximum value of S1(n,k) where n == 100:\n";
integer max = 0;
for (int k = 0; k <= 100; ++k)
max = std::max(max, s1.get(100, k));
std::cout << max << '\n';
return 0;
}</lang>
- Output:
Unsigned Stirling numbers of the first kind: n/k 0 1 2 3 4 5 6 7 8 9 10 11 12 0 1 1 0 1 2 0 1 1 3 0 2 3 1 4 0 6 11 6 1 5 0 24 50 35 10 1 6 0 120 274 225 85 15 1 7 0 720 1764 1624 735 175 21 1 8 0 5040 13068 13132 6769 1960 322 28 1 9 0 40320 109584 118124 67284 22449 4536 546 36 1 10 0 362880 1026576 1172700 723680 269325 63273 9450 870 45 1 11 0 3628800 10628640 12753576 8409500 3416930 902055 157773 18150 1320 55 1 12 0 39916800 120543840 150917976 105258076 45995730 13339535 2637558 357423 32670 1925 66 1 Maximum value of S1(n,k) where n == 100: 19710908747055261109287881673376044669240511161402863823515728791076863288440277983854056472903481625299174865860036734731122707870406148096000000000000000000
Factor
Here we calculate a row at a time as the coefficients of the falling factorial x(x-1)(x-2)...(x-n+1) using Factor's built-in polynomial arithmetic.
For example, x(x-1)(x-2) = x3 - 3x2 + 2x. Taking the absolute values of the coefficients, the third row is (0) 2 3 1.
<lang factor>USING: arrays assocs formatting io kernel math math.polynomials math.ranges prettyprint sequences ; IN: rosetta-code.stirling-first
- stirling-row ( n -- seq )
[ { 1 } ] [
[ -1 ] dip neg [a,b) dup length 1 <array> zip
{ 0 1 } [ p* ] reduce [ abs ] map
] if-zero ;
"Unsigned Stirling numbers of the first kind:" print "n\\k" write 13 dup [ "%10d" printf ] each-integer nl
[ dup "%-2d " printf stirling-row [ "%10d" printf ] each nl ] each-integer nl
"Maximum value from 100th stirling row:" print 100 stirling-row supremum .</lang>
- Output:
Unsigned Stirling numbers of the first kind: n\k 0 1 2 3 4 5 6 7 8 9 10 11 12 0 1 1 0 1 2 0 1 1 3 0 2 3 1 4 0 6 11 6 1 5 0 24 50 35 10 1 6 0 120 274 225 85 15 1 7 0 720 1764 1624 735 175 21 1 8 0 5040 13068 13132 6769 1960 322 28 1 9 0 40320 109584 118124 67284 22449 4536 546 36 1 10 0 362880 1026576 1172700 723680 269325 63273 9450 870 45 1 11 0 3628800 10628640 12753576 8409500 3416930 902055 157773 18150 1320 55 1 12 0 39916800 120543840 150917976 105258076 45995730 13339535 2637558 357423 32670 1925 66 1 Maximum value from 100th stirling row: 19710908747055261109287881673376044669240511161402863823515728791076863288440277983854056472903481625299174865860036734731122707870406148096000000000000000000
FreeBASIC
<lang freebasic>dim as integer S1(0 to 12, 0 to 12) 'initially set with zeroes dim as ubyte n, k dim as string outstr
function padto( i as ubyte, j as integer ) as string
return wspace(i-len(str(j)))+str(j)
end function
S1(0, 0) = 1
for n = 0 to 12 'calculate table
for k = 1 to n
S1(n, k) = S1(n-1, k-1) - (n-1) * S1(n-1, k)
next k
next n
print "Signed Stirling numbers of the first kind" print outstr = " k" for k=0 to 12
outstr += padto(12, k)
next k print outstr print " n" for n = 0 to 12
outstr = padto(2, n)+" "
for k = 0 to 12
outstr += padto(12, S1(n, k))
next k
print outstr
next n</lang>
Signed Stirling numbers of the first kind k 0 1 2 3 4 5 6 7 8 9 10 11 12 n 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 2 0 -1 1 0 0 0 0 0 0 0 0 0 0 3 0 2 -3 1 0 0 0 0 0 0 0 0 0 4 0 -6 11 -6 1 0 0 0 0 0 0 0 0 5 0 24 -50 35 -10 1 0 0 0 0 0 0 0 6 0 -120 274 -225 85 -15 1 0 0 0 0 0 0 7 0 720 -1764 1624 -735 175 -21 1 0 0 0 0 0 8 0 -5040 13068 -13132 6769 -1960 322 -28 1 0 0 0 0 9 0 40320 -109584 118124 -67284 22449 -4536 546 -36 1 0 0 0 10 0 -362880 1026576 -1172700 723680 -269325 63273 -9450 870 -45 1 0 0 11 0 3628800 -10628640 12753576 -8409500 3416930 -902055 157773 -18150 1320 -55 1 0 12 0 -39916800 120543840 -150917976 105258076 -45995730 13339535 -2637558 357423 -32670 1925 -66 1
Fōrmulæ
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Go
<lang go>package main
import (
"fmt" "math/big"
)
func main() {
limit := 100
last := 12
unsigned := true
s1 := make([][]*big.Int, limit+1)
for n := 0; n <= limit; n++ {
s1[n] = make([]*big.Int, limit+1)
for k := 0; k <= limit; k++ {
s1[n][k] = new(big.Int)
}
}
s1[0][0].SetInt64(int64(1))
var t big.Int
for n := 1; n <= limit; n++ {
for k := 1; k <= n; k++ {
t.SetInt64(int64(n - 1))
t.Mul(&t, s1[n-1][k])
if unsigned {
s1[n][k].Add(s1[n-1][k-1], &t)
} else {
s1[n][k].Sub(s1[n-1][k-1], &t)
}
}
}
fmt.Println("Unsigned Stirling numbers of the first kind: S1(n, k):")
fmt.Printf("n/k")
for i := 0; i <= last; i++ {
fmt.Printf("%9d ", i)
}
fmt.Printf("\n--")
for i := 0; i <= last; i++ {
fmt.Printf("----------")
}
fmt.Println()
for n := 0; n <= last; n++ {
fmt.Printf("%2d ", n)
for k := 0; k <= n; k++ {
fmt.Printf("%9d ", s1[n][k])
}
fmt.Println()
}
fmt.Println("\nMaximum value from the S1(100, *) row:")
max := new(big.Int).Set(s1[limit][0])
for k := 1; k <= limit; k++ {
if s1[limit][k].Cmp(max) > 0 {
max.Set(s1[limit][k])
}
}
fmt.Println(max)
fmt.Printf("which has %d digits.\n", len(max.String()))
}</lang>
- Output:
Unsigned Stirling numbers of the first kind: S1(n, k): n/k 0 1 2 3 4 5 6 7 8 9 10 11 12 ------------------------------------------------------------------------------------------------------------------------------------ 0 1 1 0 1 2 0 1 1 3 0 2 3 1 4 0 6 11 6 1 5 0 24 50 35 10 1 6 0 120 274 225 85 15 1 7 0 720 1764 1624 735 175 21 1 8 0 5040 13068 13132 6769 1960 322 28 1 9 0 40320 109584 118124 67284 22449 4536 546 36 1 10 0 362880 1026576 1172700 723680 269325 63273 9450 870 45 1 11 0 3628800 10628640 12753576 8409500 3416930 902055 157773 18150 1320 55 1 12 0 39916800 120543840 150917976 105258076 45995730 13339535 2637558 357423 32670 1925 66 1 Maximum value from the S1(100, *) row: 19710908747055261109287881673376044669240511161402863823515728791076863288440277983854056472903481625299174865860036734731122707870406148096000000000000000000 which has 158 digits.
J
NB. agenda set by the test according to the definition test=: 1 i.~ (0 0&-: , 1 0&-:)@:*@:, , < s1=: 1:`0:`0:`($:&<: + (($: * [)~ <:)~)@.test s1&> table i. 13 +----+------------------------------------------------------------------------------------------+ |s1&>|0 1 2 3 4 5 6 7 8 9 10 11 12| +----+------------------------------------------------------------------------------------------+ | 0 |1 0 0 0 0 0 0 0 0 0 0 0 0| | 1 |0 1 0 0 0 0 0 0 0 0 0 0 0| | 2 |0 1 1 0 0 0 0 0 0 0 0 0 0| | 3 |0 2 3 1 0 0 0 0 0 0 0 0 0| | 4 |0 6 11 6 1 0 0 0 0 0 0 0 0| | 5 |0 24 50 35 10 1 0 0 0 0 0 0 0| | 6 |0 120 274 225 85 15 1 0 0 0 0 0 0| | 7 |0 720 1764 1624 735 175 21 1 0 0 0 0 0| | 8 |0 5040 13068 13132 6769 1960 322 28 1 0 0 0 0| | 9 |0 40320 109584 118124 67284 22449 4536 546 36 1 0 0 0| |10 |0 362880 1026576 1172700 723680 269325 63273 9450 870 45 1 0 0| |11 |0 3628800 10628640 12753576 8409500 3416930 902055 157773 18150 1320 55 1 0| |12 |0 39916800 120543840 150917976 105258076 45995730 13339535 2637558 357423 32670 1925 66 1| +----+------------------------------------------------------------------------------------------+ timespacex 's1&> table i. 13' 0.0242955 12928 NB. memoization greatly helps execution time s1M=: 1:`0:`0:`($:&<: + (($: * [)~ <:)~)@.test M. timespacex 's1M&> table i. 13' 0.000235206 30336 NB. third task >./100 s1M&x:&> i.101 19710908747055261109287881673376044669240511161402863823515728791076863288440277983854056472903481625299174865860036734731122707870406148096000000000000000000
Java
<lang java> import java.math.BigInteger; import java.util.HashMap; import java.util.Map;
public class SterlingNumbersFirstKind {
public static void main(String[] args) {
System.out.println("Unsigned Stirling numbers of the first kind:");
int max = 12;
System.out.printf("n/k");
for ( int n = 0 ; n <= max ; n++ ) {
System.out.printf("%10d", n);
}
System.out.printf("%n");
for ( int n = 0 ; n <= max ; n++ ) {
System.out.printf("%-3d", n);
for ( int k = 0 ; k <= n ; k++ ) {
System.out.printf("%10s", sterling1(n, k));
}
System.out.printf("%n");
}
System.out.println("The maximum value of S1(100, k) = ");
BigInteger previous = BigInteger.ZERO;
for ( int k = 1 ; k <= 100 ; k++ ) {
BigInteger current = sterling1(100, k);
if ( current.compareTo(previous) > 0 ) {
previous = current;
}
else {
System.out.printf("%s%n(%d digits, k = %d)%n", previous, previous.toString().length(), k-1);
break;
}
}
}
private static Map<String,BigInteger> COMPUTED = new HashMap<>();
private static final BigInteger sterling1(int n, int k) {
String key = n + "," + k;
if ( COMPUTED.containsKey(key) ) {
return COMPUTED.get(key);
}
if ( n == 0 && k == 0 ) {
return BigInteger.valueOf(1);
}
if ( n > 0 && k == 0 ) {
return BigInteger.ZERO;
}
if ( k > n ) {
return BigInteger.ZERO;
}
BigInteger result = sterling1(n-1, k-1).add(BigInteger.valueOf(n-1).multiply(sterling1(n-1, k)));
COMPUTED.put(key, result);
return result;
}
} </lang>
- Output:
Unsigned Stirling numbers of the first kind: n/k 0 1 2 3 4 5 6 7 8 9 10 11 12 0 1 1 0 1 2 0 1 1 3 0 2 3 1 4 0 6 11 6 1 5 0 24 50 35 10 1 6 0 120 274 225 85 15 1 7 0 720 1764 1624 735 175 21 1 8 0 5040 13068 13132 6769 1960 322 28 1 9 0 40320 109584 118124 67284 22449 4536 546 36 1 10 0 362880 1026576 1172700 723680 269325 63273 9450 870 45 1 11 0 3628800 10628640 12753576 8409500 3416930 902055 157773 18150 1320 55 1 12 0 39916800 120543840 150917976 105258076 45995730 13339535 2637558 357423 32670 1925 66 1 The maximum value of S1(100, k) = 19710908747055261109287881673376044669240511161402863823515728791076863288440277983854056472903481625299174865860036734731122707870406148096000000000000000000 (158 digits, k = 5)
Julia
<lang julia>using Combinatorics
const s1cache = Dict()
function stirlings1(n, k, signed::Bool=false)
if signed == true && isodd(n - k)
return -stirlings1(n, k)
elseif haskey(s1cache, Pair(n, k))
return s1cache[Pair(n, k)]
elseif n < 0
throw(DomainError(n, "n must be nonnegative"))
elseif n == k == 0
return one(n)
elseif n == 0 || k == 0
return zero(n)
elseif n == k
return one(n)
elseif k == 1
return factorial(n-1)
elseif k == n - 1
return binomial(n, 2)
elseif k == n - 2
return div((3 * n - 1) * binomial(n, 3), 4)
elseif k == n - 3
return binomial(n, 2) * binomial(n, 4)
end
ret = (n - 1) * stirlings1(n - 1, k) + stirlings1(n - 1, k - 1) s1cache[Pair(n, k)] = ret return ret
end
function printstirling1table(kmax)
println(" ", mapreduce(i -> lpad(i, 10), *, 0:kmax))
sstring(n, k) = begin i = stirlings1(n, k); lpad(k > n && i == 0 ? "" : i, 10) end
for n in 0:kmax
println(rpad(n, 2) * mapreduce(k -> sstring(n, k), *, 0:kmax))
end
end
printstirling1table(12) println("\nThe maximum for stirling1(100, _) is:\n", maximum(k-> stirlings1(BigInt(100), BigInt(k)), 1:100))
</lang>
- Output:
0 1 2 3 4 5 6 7 8 9 10 11 12 0 1 1 0 1 2 0 1 1 3 0 2 3 1 4 0 6 11 6 1 5 0 24 50 35 10 1 6 0 120 274 225 85 15 1 7 0 720 1764 1624 735 175 21 1 8 0 5040 13068 13132 6769 1960 322 28 1 9 0 40320 109584 118124 67284 22449 4536 546 36 1 10 0 362880 1026576 1172700 723680 269325 63273 9450 870 45 1 11 0 3628800 10628640 12753576 8409500 3416930 902055 157773 18150 1320 55 1 12 0 39916800 120543840 150917976 105258076 45995730 13339535 2637558 357423 32670 1925 66 1 The maximum for stirling1(100, _) is: 19710908747055261109287881673376044669240511161402863823515728791076863288440277983854056472903481625299174865860036734731122707870406148096000000000000000000
Perl
<lang perl>use strict; use warnings; use bigint; use feature 'say'; use feature 'state'; no warnings 'recursion'; use List::Util qw(max);
sub Stirling1 {
my($n, $k) = @_;
return 1 unless $n || $k;
return 0 unless $n && $k;
state %seen;
return ($seen{"{$n-1}|{$k-1}"} //= Stirling1($n-1, $k-1)) +
($seen{"{$n-1}|{$k}" } //= Stirling1($n-1, $k )) * ($n-1)
}
my $upto = 12; my $width = 1 + length max map { Stirling1($upto,$_) } 0..$upto;
say 'Unsigned Stirling1 numbers of the first kind: S1(n, k):'; print 'n\k' . sprintf "%${width}s"x(1+$upto)."\n", 0..$upto;
for my $row (0..$upto) {
printf '%-3d', $row;
printf "%${width}d", Stirling1($row, $_) for 0..$row;
print "\n";
}
say "\nMaximum value from the S1(100, *) row:"; say max map { Stirling1(100,$_) } 0..100;</lang>
- Output:
Unsigned Stirling1 numbers of the first kind: S1(n, k): n\k 0 1 2 3 4 5 6 7 8 9 10 11 12 0 1 1 0 1 2 0 1 1 3 0 2 3 1 4 0 6 11 6 1 5 0 24 50 35 10 1 6 0 120 274 225 85 15 1 7 0 720 1764 1624 735 175 21 1 8 0 5040 13068 13132 6769 1960 322 28 1 9 0 40320 109584 118124 67284 22449 4536 546 36 1 10 0 362880 1026576 1172700 723680 269325 63273 9450 870 45 1 11 0 3628800 10628640 12753576 8409500 3416930 902055 157773 18150 1320 55 1 12 0 39916800 120543840 150917976 105258076 45995730 13339535 2637558 357423 32670 1925 66 1 Maximum value from the S1(100, *) row: 19710908747055261109287881673376044669240511161402863823515728791076863288440277983854056472903481625299174865860036734731122707870406148096000000000000000000
Phix
<lang Phix>include mpfr.e
constant lim = 100,
lim1 = lim+1,
last = 12
bool unsigned = true sequence s1 = repeat(0,lim1) for n=1 to lim1 do
s1[n] = mpz_inits(lim1)
end for mpz_set_si(s1[1][1],1) mpz {t, m100} = mpz_inits(2) for n=1 to lim do
for k=1 to n do
mpz_set_si(t,n-1)
mpz_mul(t,t,s1[n][k+1])
if unsigned then
mpz_add(s1[n+1][k+1],s1[n][k],t)
else
mpz_sub(s1[n+1][k+1],s1[n][k],t)
end if
end for
end for string s = iff(unsigned?"Uns":"S") printf(1,"%signed Stirling numbers of the first kind: S1(n, k):\n n k:",s) for i=0 to last do
printf(1,"%5d ", i)
end for printf(1,"\n--- %s\n",repeat('-',last*10+5)) for n=0 to last do
printf(1,"%2d ", n)
for k=1 to n+1 do
mpfr_printf(1,"%9Zd ", s1[n+1][k])
end for
printf(1,"\n")
end for for k=1 to lim1 do
mpz s100k = s1[lim1][k]
if mpz_cmp(s100k,m100) > 0 then
mpz_set(m100,s100k)
end if
end for printf(1,"\nThe maximum S1(100,k): %s\n",shorten(mpz_get_str(m100)))</lang>
- Output:
Unsigned Stirling numbers of the first kind: S1(n, k): n k: 0 1 2 3 4 5 6 7 8 9 10 11 12 --- ----------------------------------------------------------------------------------------------------------------------------- 0 1 1 0 1 2 0 1 1 3 0 2 3 1 4 0 6 11 6 1 5 0 24 50 35 10 1 6 0 120 274 225 85 15 1 7 0 720 1764 1624 735 175 21 1 8 0 5040 13068 13132 6769 1960 322 28 1 9 0 40320 109584 118124 67284 22449 4536 546 36 1 10 0 362880 1026576 1172700 723680 269325 63273 9450 870 45 1 11 0 3628800 10628640 12753576 8409500 3416930 902055 157773 18150 1320 55 1 12 0 39916800 120543840 150917976 105258076 45995730 13339535 2637558 357423 32670 1925 66 1 The maximum S1(100,k): 1971090874705526110...6000000000000000000 (158 digits)
Prolog
<lang prolog>:- dynamic stirling1_cache/3.
stirling1(N, N, 1):-!. stirling1(_, 0, 0):-!. stirling1(N, K, 0):- K > N, !. stirling1(N, K, L):- stirling1_cache(N, K, L), !. stirling1(N, K, L):- N1 is N - 1, K1 is K - 1, stirling1(N1, K, L1), stirling1(N1, K1, L2), !, L is L2 + (N - 1) * L1, assertz(stirling1_cache(N, K, L)).
print_stirling_numbers(N):- between(1, N, K), stirling1(N, K, L), writef('%10r', [L]), fail. print_stirling_numbers(_):- nl.
print_stirling_numbers_up_to(M):- between(1, M, N), print_stirling_numbers(N), fail. print_stirling_numbers_up_to(_).
max_stirling1(N, Max):-
aggregate_all(max(L), (between(1, N, K), stirling1(N, K, L)), Max).
main:- writeln('Unsigned Stirling numbers of the first kind up to S1(12,12):'), print_stirling_numbers_up_to(12), writeln('Maximum value of S1(n,k) where n = 100:'), max_stirling1(100, M), writeln(M).</lang>
- Output:
Unsigned Stirling numbers of the first kind up to S1(12,12):
1
1 1
2 3 1
6 11 6 1
24 50 35 10 1
120 274 225 85 15 1
720 1764 1624 735 175 21 1
5040 13068 13132 6769 1960 322 28 1
40320 109584 118124 67284 22449 4536 546 36 1
362880 1026576 1172700 723680 269325 63273 9450 870 45 1
3628800 10628640 12753576 8409500 3416930 902055 157773 18150 1320 55 1
39916800 120543840 150917976 105258076 45995730 13339535 2637558 357423 32670 1925 66 1
Maximum value of S1(n,k) where n = 100:
19710908747055261109287881673376044669240511161402863823515728791076863288440277983854056472903481625299174865860036734731122707870406148096000000000000000000
Python
<lang Python> computed = {}
def sterling1(n, k): key = str(n) + "," + str(k)
if key in computed.keys(): return computed[key] if n == k == 0: return 1 if n > 0 and k == 0: return 0 if k > n: return 0 result = sterling1(n - 1, k - 1) + (n - 1) * sterling1(n - 1, k) computed[key] = result return result
print("Unsigned Stirling numbers of the first kind:") MAX = 12 print("n/k".ljust(10), end="") for n in range(MAX + 1): print(str(n).rjust(10), end="") print() for n in range(MAX + 1): print(str(n).ljust(10), end="") for k in range(n + 1): print(str(sterling1(n, k)).rjust(10), end="") print() print("The maximum value of S1(100, k) = ") previous = 0 for k in range(1, 100 + 1): current = sterling1(100, k) if current > previous: previous = current else: print("{0}\n({1} digits, k = {2})\n".format(previous, len(str(previous)), k - 1)) break </lang>
- Output:
Unsigned Stirling numbers of the first kind: n/k 0 1 2 3 4 5 6 7 8 9 10 11 12 0 1 1 0 1 2 0 1 1 3 0 2 3 1 4 0 6 11 6 1 5 0 24 50 35 10 1 6 0 120 274 225 85 15 1 7 0 720 1764 1624 735 175 21 1 8 0 5040 13068 13132 6769 1960 322 28 1 9 0 40320 109584 118124 67284 22449 4536 546 36 1 10 0 362880 1026576 1172700 723680 269325 63273 9450 870 45 1 11 0 3628800 10628640 12753576 8409500 3416930 902055 157773 18150 1320 55 1 12 0 39916800 120543840 150917976 105258076 45995730 13339535 2637558 357423 32670 1925 66 1 The maximum value of S1(100, k) = 19710908747055261109287881673376044669240511161402863823515728791076863288440277983854056472903481625299174865860036734731122707870406148096000000000000000000 (158 digits, k = 5)
Raku
(formerly Perl 6)
<lang perl6>sub Stirling1 (Int \n, Int \k) {
return 1 unless n || k;
return 0 unless n && k;
state %seen;
(%seen{"{n - 1}|{k - 1}"} //= Stirling1(n - 1, k - 1)) +
(n - 1) * (%seen{"{n - 1}|{k}"} //= Stirling1(n - 1, k))
}
my $upto = 12;
my $mx = (1..^$upto).map( { Stirling1($upto, $_) } ).max.chars;
put 'Unsigned Stirling numbers of the first kind: S1(n, k):'; put 'n\k', (0..$upto)».fmt: "%{$mx}d";
for 0..$upto -> $row {
$row.fmt('%-3d').print;
put (0..$row).map( { Stirling1($row, $_) } )».fmt: "%{$mx}d";
}
say "\nMaximum value from the S1(100, *) row:"; say (^100).map( { Stirling1 100, $_ } ).max;</lang>
- Output:
Unsigned Stirling numbers of the first kind: S1(n, k): n\k 0 1 2 3 4 5 6 7 8 9 10 11 12 0 1 1 0 1 2 0 1 1 3 0 2 3 1 4 0 6 11 6 1 5 0 24 50 35 10 1 6 0 120 274 225 85 15 1 7 0 720 1764 1624 735 175 21 1 8 0 5040 13068 13132 6769 1960 322 28 1 9 0 40320 109584 118124 67284 22449 4536 546 36 1 10 0 362880 1026576 1172700 723680 269325 63273 9450 870 45 1 11 0 3628800 10628640 12753576 8409500 3416930 902055 157773 18150 1320 55 1 12 0 39916800 120543840 150917976 105258076 45995730 13339535 2637558 357423 32670 1925 66 1 Maximum value from the S1(100, *) row: 19710908747055261109287881673376044669240511161402863823515728791076863288440277983854056472903481625299174865860036734731122707870406148096000000000000000000
REXX
Some extra code was added to minimize the displaying of the column widths. <lang rexx>/*REXX program to compute and display (unsigned) Stirling numbers of the first kind.*/ parse arg lim . /*obtain optional argument from the CL.*/ if lim== | lim=="," then lim= 12 /*Not specified? Then use the default.*/ olim= lim /*save the original value of LIM. */ lim= abs(lim) /*only use the absolute value of LIM. */ numeric digits max(9, 2*lim) /*(over) specify maximum number in grid*/ @.=; @.0.0= 1 /*define the (0, 0)th value in grid*/
do n=0 for lim+1 /* [↓] initialize some values " " */
if n>0 then @.n.0 = 0 /*assign value if N > zero. */
do k=n+1 to lim
@.n.k = 0 /*zero some values in grid if K > N. */
end /*k*/
end /*n*/
max#.= 0 /* [↓] calculate values for the grid. */
do n=1 for lim; nm= n - 1
do k=1 for lim; km= k - 1
@.n.k = @.nm.km + nm * @.nm.k /*calculate an unsigned number in grid.*/
max#.k= max(max#.k, @.n.k) /*find the maximum value " " */
max#.b= max(max#.b, @.n.k) /*find the maximum value for all rows. */
end /*k*/
end /*n*/
do k=1 for lim /*find max column width for each column*/
max#.a= max#.a + length(max#.k)
end /*k*/
/* [↓] only show the maximum value ? */
w= length(max#.b) /*calculate max width of all numbers. */ if olim<0 then do; say 'The maximum value (which has ' w " decimal digits):"
say max#.b /*display maximum number in the grid. */
exit /*stick a fork in it, we're all done. */
end
wi= max(3, length(lim+1) ) /*the maximum width of the grid's index*/ say 'row' center('columns', max(9, max#.a + lim), '═') /*display header of the grid.*/
do r=0 for lim+1; $= /* [↓] display the grid to the term. */
do c=0 for lim+1 until c>=r /*build a row of grid, 1 col at a time.*/
$= $ right(@.r.c, length(max#.c) ) /*append a column to a row of the grid.*/
end /*c*/
say right(r,wi) strip(substr($,2), 'T') /*display a single row of the grid. */
end /*r*/ /*stick a fork in it, we're all done. */</lang>
- output when using the default input:
row ════════════════════════════════════════columns═════════════════════════════════════════ 0 1 1 0 1 2 0 1 1 3 0 2 3 1 4 0 6 11 6 1 5 0 24 50 35 10 1 6 0 120 274 225 85 15 1 7 0 720 1764 1624 735 175 21 1 8 0 5040 13068 13132 6769 1960 322 28 1 9 0 40320 109584 118124 67284 22449 4536 546 36 1 10 0 362880 1026576 1172700 723680 269325 63273 9450 870 45 1 11 0 3628800 10628640 12753576 8409500 3416930 902055 157773 18150 1320 55 1 12 0 39916800 120543840 150917976 105258076 45995730 13339535 2637558 357423 32670 1925 66 1
- output when using the input of: -100
The maximum value (which has 158 decimal digits): 19710908747055261109287881673376044669240511161402863823515728791076863288440277983854056472903481625299174865860036734731122707870406148096000000000000000000
Sidef
<lang ruby>func S1(n, k) { # unsigned Stirling numbers of the first kind
stirling(n, k).abs
}
const r = (0..12)
var triangle = r.map {|n| 0..n -> map {|k| S1(n, k) } } var widths = r.map {|n| r.map {|k| (triangle[k][n] \\ 0).len }.max }
say ('n\k ', r.map {|n| "%*s" % (widths[n], n) }.join(' '))
r.each {|n|
var str = ('%-3s ' % n)
str += triangle[n].map_kv {|k,v| "%*s" % (widths[k], v) }.join(' ')
say str
}
with (100) {|n|
say "\nMaximum value from the S1(#{n}, *) row:"
say { S1(n, _) }.map(^n).max
}</lang>
- Output:
n\k 0 1 2 3 4 5 6 7 8 9 10 11 12 0 1 1 0 1 2 0 1 1 3 0 2 3 1 4 0 6 11 6 1 5 0 24 50 35 10 1 6 0 120 274 225 85 15 1 7 0 720 1764 1624 735 175 21 1 8 0 5040 13068 13132 6769 1960 322 28 1 9 0 40320 109584 118124 67284 22449 4536 546 36 1 10 0 362880 1026576 1172700 723680 269325 63273 9450 870 45 1 11 0 3628800 10628640 12753576 8409500 3416930 902055 157773 18150 1320 55 1 12 0 39916800 120543840 150917976 105258076 45995730 13339535 2637558 357423 32670 1925 66 1 Maximum value from the S1(100, *) row: 19710908747055261109287881673376044669240511161402863823515728791076863288440277983854056472903481625299174865860036734731122707870406148096000000000000000000
Alternatively, the S1(n,k) function can be defined as: <lang ruby>func S1((0), (0)) { 1 } func S1(_, (0)) { 0 } func S1((0), _) { 0 } func S1(n, k) is cached { S1(n-1, k-1) + (n-1)*S1(n-1, k) }</lang>
Tcl
This computes the unsigned Stirling numbers of the first kind. Inspired by Stirling_numbers_of_the_second_kind#Tcl, hence similar to #Java. <lang Tcl>proc US1 {n k} {
if {$k == 0} {
return [expr {$n == 0}]
}
if {$n < $k} {
return 0
}
if {$n == $k} {
return 1
}
set nk [list $n $k]
if {[info exists ::US1cache($nk)]} {
return $::US1cache($nk)
}
set n1 [expr {$n - 1}]
set k1 [expr {$k - 1}]
set r [expr {($n1 * [US1 $n1 $k]) + [US1 $n1 $k1]}]
set ::US1cache($nk) $r
}
proc main {} {
puts "Unsigned Stirling numbers of the first kind:"
set max 12 ;# last table line to print
set L 9 ;# space to use for 1 number
puts -nonewline "n\\k"
for {set n 0} {$n <= $max} {incr n} {
puts -nonewline " [format %${L}d $n]"
}
puts ""
for {set n 0} {$n <= $max} {incr n} {
puts -nonewline [format %-3d $n]
for {set k 0} {$k <= $n} {incr k} {
puts -nonewline " [format %${L}s [US1 $n $k]]"
}
puts ""
}
puts "The maximum value of US1(100, k) = "
set previous 0
for {set k 1} {$k <= 100} {incr k} {
set current [US1 100 $k]
if {$current > $previous} {
set previous $current
} else {
puts $previous
puts "([string length $previous] digits, k = [expr {$k-1}])"
break
}
}
} main</lang>
- Output:
Unsigned Stirling numbers of the first kind: n\k 0 1 2 3 4 5 6 7 8 9 10 11 12 0 1 1 0 1 2 0 1 1 3 0 2 3 1 4 0 6 11 6 1 5 0 24 50 35 10 1 6 0 120 274 225 85 15 1 7 0 720 1764 1624 735 175 21 1 8 0 5040 13068 13132 6769 1960 322 28 1 9 0 40320 109584 118124 67284 22449 4536 546 36 1 10 0 362880 1026576 1172700 723680 269325 63273 9450 870 45 1 11 0 3628800 10628640 12753576 8409500 3416930 902055 157773 18150 1320 55 1 12 0 39916800 120543840 150917976 105258076 45995730 13339535 2637558 357423 32670 1925 66 1 The maximum value of US1(100, k) = 19710908747055261109287881673376044669240511161402863823515728791076863288440277983854056472903481625299174865860036734731122707870406148096000000000000000000 (158 digits, k = 5)
zkl
<lang zkl>fcn stirling1(n,k){
var seen=Dictionary(); // cache for recursion
if(n==k==0) return(1);
if(n==0 or k==0) return(0);
z1,z2 := "%d,%d".fmt(n-1,k-1), "%d,%d".fmt(n-1,k);
if(Void==(s1 := seen.find(z1))){ s1 = seen[z1] = stirling1(n - 1, k - 1) }
if(Void==(s2 := seen.find(z2))){ s2 = seen[z2] = stirling1(n - 1, k) }
(n - 1)*s2 + s1; // n is first to cast to BigInt (if using BigInts)
}</lang> <lang zkl>// calculate entire table (cached), find max, find num digits in max N,mx := 12, [1..N].apply(fcn(n){ [1..n].apply(stirling1.fp(n)) : (0).max(_) }) : (0).max(_); fmt:="%%%dd".fmt("%d".fmt(mx.numDigits + 1)).fmt; // "%9d".fmt println("Unsigned Stirling numbers of the first kind: S1(n, k):"); println("n\\k",[0..N].pump(String,fmt)); foreach row in ([0..N]){
println("%3d".fmt(row), [0..row].pump(String, stirling1.fp(row), fmt));
}</lang>
- Output:
Unsigned Stirling numbers of the first kind: S1(n, k): n\k 0 1 2 3 4 5 6 7 8 9 10 11 12 0 1 1 0 1 2 0 1 1 3 0 2 3 1 4 0 6 11 6 1 5 0 24 50 35 10 1 6 0 120 274 225 85 15 1 7 0 720 1764 1624 735 175 21 1 8 0 5040 13068 13132 6769 1960 322 28 1 9 0 40320 109584 118124 67284 22449 4536 546 36 1 10 0 362880 1026576 1172700 723680 269325 63273 9450 870 45 1 11 0 3628800 10628640 12753576 8409500 3416930 902055 157773 18150 1320 55 1 12 0 39916800 120543840 150917976 105258076 45995730 13339535 2637558 357423 32670 1925 66 1
GNU Multiple Precision Arithmetic Library
<lang zkl>var [const] BI=Import("zklBigNum"); // libGMP N=100; println("Maximum value from the S1(%d, *) row:".fmt(N)); [1..N].apply(stirling1.fp(BI(N)))
.reduce(fcn(m,n){ m.max(n) }).println();</lang>
- Output:
Maximum value from the S1(100, *) row: 19710908747055261109287881673376044669240511161402863823515728791076863288440277983854056472903481625299174865860036734731122707870406148096000000000000000000