OFFSET
0,6
COMMENTS
This array gives in row n>=1 the multinomial numbers (call them M_0 numbers) m!/product((a_j)!,j=1..n) with the exponents of the partitions of n with number of parts m:=sum(a_j,j=1..n), given in the Abramowitz-Stegun order. See p. 831 of the given reference. See also the arrays for the M_1, M_2 and M_3 multinomial numbers A036038, A036039 and A036040 (or A080575).
For a signed version see A111786.
These M_0 multinomial numbers give the number of compositions of n >= 1 with parts corresponding to the partitions of n (in A-St order). See an n = 5 example below. The triangle with the summed entries of like number of parts m is A007318(n-1, m-1) (Pascal). - Wolfdieter Lang, Jan 29 2021
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..2713 (rows 0..20)
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1972.
Wolfdieter Lang, First 10 rows and more.
FORMULA
If the n-th partition is P, a(n) is the multinomial coefficient of the signature of P. - Franklin T. Adams-Watters, May 30 2006
EXAMPLE
Table starts:
[1]
[1]
[1, 1]
[1, 2, 1]
[1, 2, 1, 3, 1]
[1, 2, 2, 3, 3, 4, 1]
[1, 2, 2, 1, 3, 6, 1, 4, 6, 5, 1]
[1, 2, 2, 2, 3, 6, 3, 3, 4, 12, 4, 5, 10, 6, 1]
.
T(5,6) = 4 because there are four multisets using the first four digits {0,1,2,3}: 32100, 32110, 32210 and 33210
T(5,6) = 4 because there are 4 compositions of 5 that can be formed from the partition 2+1+1+1. - Geoffrey Critzer, May 19 2013
These 4 compositions 2+1+1+1, 1+2+1+1, 1+1+2+1 and 1+1+1+2 of 5 correspond to the 4 set partitions of [5] :={1,2,3,4,5}, with 4 blocks of consecutive numbers, namely {1,2},{3},{4},{5} and {1},{2,3},{4},{5} and {1},{2},{3,4},{5} and {1},{2},{3},{4,5}. - Wolfdieter Lang, May 30 2018
MAPLE
nmax:=9: with(combinat): for n from 1 to nmax do P(n):=sort(partition(n)): for r from 1 to numbpart(n) do B(r):=P(n)[r] od: for m from 1 to numbpart(n) do s:=0: j:=0: while s<n do j:=j+1: s:=s+B(m)[j]: x(j):=B(m)[j]: end do; jmax:=j; for r from 1 to n do q(r):=0 od: for r from 1 to n do for j from 1 to jmax do if x(j)=r then q(r):=q(r)+1 fi: od: od: A036040(n, m) := (add(q(t), t=1..n))!/(mul(q(t)!, t=1..n)); od: od: seq(seq(A036040(n, m), m=1..numbpart(n)), n=1..nmax); # Johannes W. Meijer, Jul 14 2016
PROG
(SageMath) from collections import Counter
def ASPartitions(n, k):
Q = [p.to_list() for p in Partitions(n, length=k)]
for q in Q: q.reverse()
return sorted(Q)
def A048996_row(n):
h = lambda p: product(map(factorial, Counter(p).values()))
return [factorial(len(p))//h(p) for k in (0..n) for p in ASPartitions(n, k)]
for n in (1..10): print(A048996_row(n)) # Peter Luschny, Nov 02 2019 [corrected on notice from Sean A. Irvine, Apr 30 2022]
(PARI)
C(sig)={my(S=Set(sig)); (#sig)!/prod(k=1, #S, (#select(t->t==S[k], sig))!)}
Row(n)={apply(C, [Vecrev(p) | p<-partitions(n)])}
{ for(n=0, 7, print(Row(n))) } \\ Andrew Howroyd, Oct 18 2020
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
EXTENSIONS
More terms from Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Jun 17 2001
a(0)=1 prepended by Andrew Howroyd, Oct 19 2020
STATUS
approved