A227955 - OEIS

%I #31 Mar 14 2015 01:02:20

%S 1,2,6,4,30,12,8,210,60,36,24,16,2310,420,180,120,72,48,32,30030,4620,

%T 1260,900,840,360,216,240,144,96,64,510510,60060,13860,6300,9240,2520,

%U 1800,1080,1680,720,432,480,288,192,128,9699690,1021020,180180,69300,44100

%N Triangle read by rows, T(n, k) = prime(1)^p(k,1)*...*prime(n)^p(k,n) where p(k,j) is the j-th part of the k-th partition of n. The partitions of n are ordered in reversed lexicographic order read from left-to-right, starting with [1,1,...1] going down to [n].

%C The sequence can be seen as an encoding of Young's lattice (see the links).

%C The ordering of Young's lattice is such that for two Young diagrams s, t, we have s <= t if and only if the Young diagram of s fits entirely inside the Young diagram of t (when the two diagrams are arranged so their lower-left corners coincide.) This order translates to our encoding as the divisibility relation. The number corresponding to s divides the number corresponding to t if and only if s <= t.

%C The partition corresponding to a number can be recovered as the exponents of the primes in the prime factorization of the number.

%H Peter Luschny, <a href="/A227955/b227955.txt">Rows n = 0..25, flattened</a>

%H Peter Luschny, <a href="/A227955/a227955.jpg">Young's lattice (diagram)</a>

%H Peter Luschny, <a href="http://oeis.org/wiki/User:Peter_Luschny/IntegerPartitionTrees">Integer partition trees</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Young%27s_lattice">Young's lattice</a>

%e For instance the partitions of 4 are ordered [1,1,1,1], [2,1,1,0], [2,2,0,0], [3,1,0,0], [4,0,0,0]. Consider the partition P = (3,2,1,1) written as a Young diagram (in French notation):

%e     [ ]

%e     [ ]

%e     [ ][ ]

%e     [ ][ ][ ]

%e Next replace the boxes at the bottom line by the sequence of primes and write the number of boxes in the same column as exponents; then multiply. 2^4*3^2*5^1 = 720. 720 will appear in line 7 of the triangle (because P is a partition of 7) at position 10 (because the sequence of exponents [4, 2, 1] is the 10th partition in the order of partitions which we assume).

%e [0]     1,

%e [1]     2,

%e [2]     6,    4,

%e [3]    30,   12,    8,

%e [4]   210,   60,   36,  24,  16,

%e [5]  2310,  420,  180, 120,  72,  48,  32,

%e [6] 30030, 4620, 1260, 900, 840, 360, 216, 240, 144, 96, 64.

%p with(combinat):

%p A227955_row := proc(n) local e, w, p;

%p p := [seq(ithprime(i), i=1..n)];

%p w := e -> mul(p[i]^e[nops(e)-i+1], i=1..nops(e));

%p seq(w(e), e = partition(n)) end:

%p seq(print(A227955_row(i)), i=0..8);

%o (Sage)

%o def A227955_row(n):

%o     L = []

%o     P = primes_first_n(n)

%o     for p in Partitions(n):

%o         L.append(mul(P[i]^p[i] for i in range(len(p))))

%o     return L[::-1]

%o for n in (0..8): A227955_row(n)

%Y Reversed rows: A036035, row sums: A074140.

%K nonn,tabf

%O 0,2

%A _Peter Luschny_, Aug 01 2013