A382041 Triangle read by rows: T(n, k) is the number of partitions of n with at most k parts where 0 <= k <= n, and each part is one of four kinds.

1, 0, 4, 0, 4, 14, 0, 4, 20, 40, 0, 4, 30, 70, 105, 0, 4, 36, 116, 196, 252, 0, 4, 46, 170, 350, 490, 574, 0, 4, 52, 236, 556, 896, 1120, 1240, 0, 4, 62, 310, 845, 1505, 2079, 2415, 2580, 0, 4, 68, 400, 1200, 2400, 3584, 4480, 4960, 5180, 0, 4, 78, 494, 1670, 3626, 5910, 7842, 9162, 9822, 10108

Offset: 0

Author

Peter Dolland, Mar 12 2025

Keywords

• nonn
• tabl

Comments

Two unrestricted unary predicates on the parts set result in four kinds: The intersection, the both differences and the complement of the union.

The 1-kind case is Euler's table A026820.

The 2-kind case is A381895.

The 3-kind case is A382025.

Examples

			Triangle starts:
  0 : [1]
  1 : [0, 4]
  2 : [0, 4, 14]
  3 : [0, 4, 20,  40]
  4 : [0, 4, 30,  70,  105]
  5 : [0, 4, 36, 116,  196,  252]
  6 : [0, 4, 46, 170,  350,  490,  574]
  7 : [0, 4, 52, 236,  556,  896, 1120, 1240]
  8 : [0, 4, 62, 310,  845, 1505, 2079, 2415, 2580]
  9 : [0, 4, 68, 400, 1200, 2400, 3584, 4480, 4960, 5180]
 10 : [0, 4, 78, 494, 1670, 3626, 5910, 7842, 9162, 9822, 10108]
 ...
		

Crossrefs

Main diagonal gives A023003.

Cf. A026820, A381895, A382025.

Programs

Python

from sympy import binomial
from sympy.utilities.iterables import partitions
from sympy.combinatorics.partitions import IntegerPartition
kinds = 4 - 1   # the number of part kinds - 1
def a382041_row( n):
    if n == 0 : return [1]
    t = list( [0] * n)
    for p in partitions( n):
        p = IntegerPartition( p).as_dict()
        fact = 1
        s = 0
        for k in p :
            s += p[k]
            fact *= binomial( kinds + p[k], kinds)
        if s > 0 :
            t[s - 1] += fact
    for i in range( n - 1):
        t[i+1] += t[i]
    return [0] + t