A122980
Number of distributive sublattices of the lattice of k-tuples less than the n-th partition (in Mathematica order), that include the maximum element.
Original entry on oeis.org
2, 4, 7, 8, 21, 45, 16, 58, 84, 200, 500, 32, 152, 293, 748, 1184, 3220
Offset: 1
For a(5), partition [2,1], the lattice consists of the 6 pairs (i,j) where 0<=i<=2 and 0<=j<=1, with (i,j) <= (i',j') iff i<=i' and j<=j'. {(2,1), (2,0), (0,1), (0,0)} is one distributive sublattice.
A122981
Number of distributive sublattices of the lattice of k-tuples less than the n-th partition (in Abramowitz and Stegun order).
Original entry on oeis.org
3, 7, 12, 15, 37, 73, 31, 103, 146, 319, 731, 63, 271, 505, 1191, 1833, 4618
Offset: 1
For a(5), partition [2,1], the lattice consists of the 6 pairs (i,j) where 0<=i<=2 and 0<=j<=1, with (i,j) <= (i',j') iff i<=i' and j<=j'. {(2,1), (2,0), (0,1), (0,0)} is one distributive sublattice.
- M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
A122982
Number of distributive sublattices of the lattice of k-tuples less than the n-th partition (in Mathematica order).
Original entry on oeis.org
3, 7, 12, 15, 37, 73, 31, 103, 146, 319, 731, 63, 271, 505, 1191, 1833, 4618
Offset: 1
For a(5), partition [2,1], the lattice consists of the 6 pairs (i,j) where 0<=i<=2 and 0<=j<=1, with (i,j) <= (i',j') iff i<=i' and j<=j'. {(2,1), (2,0), (0,1), (0,0)} is one distributive sublattice.
A238957
The number of nodes at even level in divisor lattice in graded colexicographic order.
Original entry on oeis.org
1, 1, 2, 2, 2, 3, 4, 3, 4, 5, 6, 8, 3, 5, 6, 8, 9, 12, 16, 4, 6, 8, 8, 10, 12, 14, 16, 18, 24, 32, 4, 7, 9, 10, 12, 15, 16, 18, 20, 24, 27, 32, 36, 48, 64, 5, 8, 11, 12, 13, 14, 18, 20, 23, 24, 24, 30, 32, 36, 41, 40, 48, 54, 64, 72, 96, 128
Offset: 0
Triangle T(n,k) begins:
1;
1;
2, 2;
2, 3, 4;
3, 4, 5, 6, 8;
3, 5, 6, 8, 9, 12, 16;
4, 6, 8, 8, 10, 12, 14, 16, 18, 24, 32;
...
Cf.
A038548 in graded colexicographic order.
-
\\ here b(n) is A038548.
b(n)={ceil(numdiv(n)/2)}
N(sig)={prod(k=1, #sig, prime(k)^sig[k])}
Row(n)={apply(s->b(N(s)), [Vecrev(p) | p<-partitions(n)])}
{ for(n=0, 8, print(Row(n))) } \\ Andrew Howroyd, Apr 01 2020
Offset changed and terms a(50) and beyond from
Andrew Howroyd, Apr 01 2020
A238958
The number of nodes at odd level in divisor lattice in graded colexicographic order.
Original entry on oeis.org
0, 1, 1, 2, 2, 3, 4, 2, 4, 4, 6, 8, 3, 5, 6, 8, 9, 12, 16, 3, 6, 7, 8, 10, 12, 13, 16, 18, 24, 32, 4, 7, 9, 10, 12, 15, 16, 18, 20, 24, 27, 32, 36, 48, 64, 4, 8, 10, 12, 12, 14, 18, 20, 22, 24, 24, 30, 32, 36, 40, 40, 48, 54, 64, 72, 96, 128
Offset: 0
Triangle T(n,k) begins:
0;
1;
1, 2;
2, 3, 4;
2, 4, 4, 6, 8;
3, 5, 6, 8, 9, 12, 16;
3, 6, 7, 8, 10, 12, 13, 16, 18, 24, 32;
...
Cf.
A056924 in graded colexicographic order.
-
\\ here b(n) is A056924.
b(n)={numdiv(n)\2}
N(sig)={prod(k=1, #sig, prime(k)^sig[k])}
Row(n)={apply(s->b(N(s)), [Vecrev(p) | p<-partitions(n)])}
{ for(n=0, 8, print(Row(n))) } \\ Andrew Howroyd, Apr 01 2020
Offset changed and terms a(50) and beyond from
Andrew Howroyd, Apr 01 2020
A079308
For a partition P of a positive integer, let f(P) be the product of k+1, over all parts k in P. Let a(n,r) be the sum of f(P) over all partitions P of n with smallest part r. Sequence gives table of a(n,r) for 1 <= r <= n, in the order a(1,1); a(2,1), a(2,2); a(3,1), a(3,2), a(3,3); ...
Original entry on oeis.org
2, 4, 3, 14, 0, 4, 36, 9, 0, 5, 100, 12, 0, 0, 6, 236, 42, 16, 0, 0, 7, 602, 54, 20, 0, 0, 0, 8, 1368, 195, 24, 25, 0, 0, 0, 9, 3242, 246, 92, 30, 0, 0, 0, 0, 10, 7240, 759, 112, 35, 36, 0, 0, 0, 0, 11, 16386, 1134, 232, 40, 42, 0, 0, 0, 0, 0, 12, 35692, 2859, 528, 170, 48, 49, 0, 0, 0, 0, 0, 13
Offset: 1
The partitions with minimal part 3 begin 3, 3+3, 4+3, 5+3, 6+3, 3+3+3, ... which yield the following values of f: 4, 16, 20, 24, 28, 64, ... therefore the 3rd column of our table begins 4,0,0,16,20,24,(28+64)=92,...
Triangle a(n,r) begins:
: 2;
: 4, 3;
: 14, 0, 4;
: 36, 9, 0, 5;
: 100, 12, 0, 0, 6;
: 236, 42, 16, 0, 0, 7;
: 602, 54, 20, 0, 0, 0, 8;
: 1368, 195, 24, 25, 0, 0, 0, 9;
: 3242, 246, 92, 30, 0, 0, 0, 0, 10;
: 7240, 759, 112, 35, 36, 0, 0, 0, 0, 11;
-
b:= proc(n, k) option remember; `if`(n=0, 1,
`if`(k>n, 0, b(n, k+1) +(k+1)*b(n-k, k)))
end:
a:= (n, k)-> b(n,k)-b(n, k+1):
seq(seq(a(n, k), k=1..n), n=1..12); # Alois P. Heinz, May 22 2015
-
a[n_, r_] := Which[r>n, 0, r==n, n+1, True, a[n, r]=(r+1)Sum[a[n-r, s], {s, r, n-r}]]; Flatten[Table[a[n, r], {n, 1, 12}, {r, 1, n}]]
A078436
Triangle read by rows in which n-th row counts multisets associated with hook partitions.
Original entry on oeis.org
1, 2, 0, 3, 4, 0, 4, 6, 8, 0, 5, 8, 12, 16, 0, 6, 10, 16, 24, 32, 0, 7, 12, 20, 32, 48, 64, 0, 8, 14, 24, 40, 64, 96, 128, 0, 9, 16, 28, 48, 80, 128, 192, 256, 0, 10, 18, 32, 56, 96, 160, 256, 384, 512, 0, 11, 20, 36, 64, 112, 192, 320, 512, 768, 1024, 0, 12, 22, 40, 72, 128
Offset: 1
Triangle begins 1; 2,0; 3,4,0; 4,6,8,0; 5,8,12,16,0; ...
a(13) = 12 because we find 1 + 3 + 4 + 3 + 1 multisets of type 21^(n-2): they are 4; 14,24,34; 114,124,134,234; 1124,1134,1234; and 11234
A122453
Triangular array related to A122402 as A079025 relates to A122172; sums A079139 values associated with cyclic partitions.
Original entry on oeis.org
1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 3, 2, 2, 3, 4, 4, 3, 2, 4, 8, 13, 15, 13, 8, 4, 4, 8, 13, 16, 16, 13, 8, 4, 7, 17, 32, 45, 51, 45, 32, 17, 7, 8, 20, 38, 56, 67, 67, 56, 38, 20, 8, 12, 34, 72, 117, 156, 171, 156, 117, 72, 34, 12, 14, 41, 88, 147, 203, 237, 237, 203, 147, 88, 41
Offset: 0
The table begins:
1
0 0
1 1 1
1 1 1 1
2 3 4 3 2
2 3 4 4 3 2
4 8 13 15 13 8 4
4 8 13 16 16 13 8 4
7 17 32 45 51 45 32 17 7
8 20 38 56 67 67 56 38 20 8
12 34 72 117 156 171 156 117 72 34 12
14 41 88 147 203 237 237 203 147 88 41 14
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