login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A048996 Irregular triangle read by rows. Preferred multisets: numbers refining A007318 using format described in A036038. 35
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, 1, 2, 2, 2, 1, 3, 6, 6, 3, 3, 4, 12, 6, 12, 1, 5, 20, 10, 6, 15, 7, 1, 1, 2, 2, 2, 2, 3, 6, 6, 3, 3, 6, 1, 4, 12, 12, 12, 12, 4, 5, 20, 10, 30, 5, 6, 30, 20, 7, 21, 8, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,5

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.

LINKS

Table of n, a(n) for n=1..96.

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

T(n,k) = A036040(n,k) * Factorial(A036043(n,k)) / A036038(n,k) = A049019(n,k) / A036038(n,k).

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, 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 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 Partitions(n, length=k)]

for n in (1..10): print(A048996_row(n)) # Peter Luschny, Nov 02 2019

CROSSREFS

Cf. A000670, A007318, A036038, A019538, A115621, A000079 (row sums), A000041 (row lengths).

Sequence in context: A165357 A210961 A250007 * A111786 A072811 A296559

Adjacent sequences:  A048993 A048994 A048995 * A048997 A048998 A048999

KEYWORD

nonn,tabf

AUTHOR

Alford Arnold

EXTENSIONS

More terms from Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Jun 17 2001

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified May 24 19:14 EDT 2020. Contains 334580 sequences. (Running on oeis4.)