mathtools/internal/lib/partitions.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] = p_{k-1}(i) for a given k, nextPartition updates the slice so vals[i] = p_k(i) using the property that p_k(n) = p_k(n-k) + p_{k-1}(n)
func nextPartition(k int, vals []*big.Int) {
	for i := 1; i < len(vals); i++ {
		if i-k >= 0 {
			vals[i].Add(vals[i], vals[i-k])
		}
	}
}

/*
Partitions computes the number of ways to add to n with k or fewer terms.
*/
func Partitions(n, k int) *big.Int {
	if n < 0 {
		return big.NewInt(0)
	}

	if k > n {
		k = n
	}

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

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

	return vals[n]
}