mathtools/internal/lib/stirling.go

/*
Copyright © 2025 filifa

This program is free software: you can redistribute it and/or modify
it under the terms of the GNU General Public License as published by
the Free Software Foundation, either version 3 of the License, or
(at your option) any later version.

This program is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
GNU General Public License for more details.

You should have received a copy of the GNU General Public License
along with this program. If not, see <http://www.gnu.org/licenses/>.
*/
package lib

import (
	"math/big"
)

// given a slice where vals[i] = Stirling1(i+k-1, k-1) for a given k, nextStirling1 updates the slice so vals[i] = Stirling1(i+k, k) using the property that Stirling1(n, k) = -(n-1)*Stirling1(n-1, k) + Stirling1(n-1, k-1)
func nextStirling1(k int, vals []*big.Int) {
	for i := 1; i < len(vals); i++ {
		n := int64(k + i - 1)
		v := big.NewInt(-n)
		v.Mul(v, vals[i-1])
		vals[i].Add(vals[i], v)
	}
}

func Stirling1(n, k int) *big.Int {
	if k > n {
		return big.NewInt(0)
	}

	vals := make([]*big.Int, n-k+1)
	for i := range vals {
		vals[i] = big.NewInt(0)
	}
	vals[0] = big.NewInt(1)

	for i := 1; i <= k; i++ {
		nextStirling1(i, vals)
	}

	return vals[n-k]
}

// given a slice where vals[i] = Stirling2(i+k-1, k-1) for a given k, nextStirling2 updates the slice so vals[i] = Stirling2(i+k, k) using the property that Stirling2(n, k) = k*Stirling2(n-1, k) + Stirling2(n-1, k-1)
func nextStirling2(k int64, vals []*big.Int) {
	for i := 1; i < len(vals); i++ {
		v := big.NewInt(k)
		v.Mul(v, vals[i-1])
		vals[i].Add(vals[i], v)
	}
}

func Stirling2(n, k int) *big.Int {
	if k > n {
		return big.NewInt(0)
	}

	vals := make([]*big.Int, n-k+1)
	for i := range vals {
		vals[i] = big.NewInt(0)
	}
	vals[0] = big.NewInt(1)

	for i := 1; i <= k; i++ {
		nextStirling2(int64(i), vals)
	}

	return vals[n-k]
}