A275069 -id:A275069 - cp's OEIS frontend

A124419 Number of partitions of the set {1,2,...n} having no blocks that contain both odd and even entries.

1, 1, 1, 2, 4, 10, 25, 75, 225, 780, 2704, 10556, 41209, 178031, 769129, 3630780, 17139600, 87548580, 447195609, 2452523325, 13450200625, 78697155750, 460457244900, 2859220516290, 17754399678409, 116482516809889, 764214897046969, 5277304280371714

Offset: 0

Emeric Deutsch, Oct 31 2006

nonn

			a(4) = 4 because we have 13|24, 1|24|3, 13|2|4 and 1|2|3|4.

Alois P. Heinz, Table of n, a(n) for n = 0..500
A. Dzhumadil’daev and D. Yeliussizov, Path decompositions of digraphs and their applications to Weyl algebra, arXiv preprint arXiv:1408.6764v1, 2014. [Version 1 contained many references to the OEIS, which were removed in Version 2. - _N. J. A. Sloane_, Mar 28 2015]
Askar Dzhumadil’daev and Damir Yeliussizov, Walks, partitions, and normal ordering, Electronic Journal of Combinatorics, 22(4) (2015), #P4.10.

Cf. A000110, A088911, A124418, A124420, A124421, A124422, A124423, A152875, A274310.
Column k=0 of A124418 and of A363493.
Column k=2 of A275069.

Q[0]:=1: for n from 1 to 30 do if n mod 2 = 1 then Q[n]:=expand(t*diff(Q[n-1],t)+x*diff(Q[n-1],s)+x*diff(Q[n-1],x)+t*Q[n-1]) else Q[n]:=expand(x*diff(Q[n-1],t)+s*diff(Q[n-1],s)+x*diff(Q[n-1],x)+s*Q[n-1]) fi od: for n from 0 to 30 do Q[n]:=Q[n] od: seq(subs({t=1,s=1,x=0},Q[n]),n=0..30);
# second Maple program:
with(combinat):
a:= n-> bell(floor(n/2))*bell(ceil(n/2)):
seq(a(n), n=0..30);  # Alois P. Heinz, Oct 23 2013

a[n_] := BellB[Floor[n/2]]*BellB[Ceiling[n/2]]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, May 20 2015, after Alois P. Heinz *)

a(n) = Q[n](1,1,0), where the polynomials Q[n]=Q[n](t,s,x) are defined by Q[0]=1; Q[n]=t*dQ[n-1]/dt + x*dQ[n-1]/ds + x*dQ[n-1]/dx + t*Q[n-1] if n is odd and Q[n]=x*dQ[n-1]/dt + s*dQ[n-1]/ds + x*dQ[n-1]/dx + s*Q[n-1] if n is even.
a(n) = A000110(floor(n/2)) * A000110(ceiling(n/2)). - Alois P. Heinz, Oct 23 2013
a(n) mod 2 = A088911(n). - Alois P. Heinz, Jun 06 2023

A275070 Number of set partitions of [n] such that i-j is a multiple of three for all i,j belonging to the same block.

Original entry on oeis.org

1, 1, 1, 1, 2, 4, 8, 20, 50, 125, 375, 1125, 3375, 11700, 40560, 140608, 548912, 2142868, 8365427, 36140293, 156133187, 674526133, 3184194060, 15031429200, 70957944000, 362451121200, 1851389821260, 9456845543523, 51863510753775, 284431392616875

Offset: 0

Alois P. Heinz, Jul 15 2016

nonn

			a(7) = 20: 147|25|36, 14|25|36|7, 147|25|3|6, 14|25|3|6|7, 147|2|36|5, 14|2|36|5|7, 147|2|3|5|6, 14|2|3|5|6|7, 17|25|36|4, 1|25|36|47, 1|25|36|4|7, 17|25|3|4|6, 1|25|3|47|6, 1|25|3|4|6|7, 17|2|36|4|5, 1|2|36|47|5, 1|2|36|4|5|7, 17|2|3|4|5|6, 1|2|3|47|5|6, 1|2|3|4|5|6|7.

Alois P. Heinz, Table of n, a(n) for n = 0..699
Wikipedia, Partition of a set

Column k=3 of A275069.

Cf. A000110.

a(n) = Product_{i=0..2} A000110(floor((n+i)/3)).

A275071 Number of set partitions of [n] such that i-j is a multiple of four for all i,j belonging to the same block.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 4, 8, 16, 40, 100, 250, 625, 1875, 5625, 16875, 50625, 175500, 608400, 2109120, 7311616, 28543424, 111429136, 435002204, 1698181681, 7336479479, 31695036961, 136928804999, 591559418641, 2792538190620, 13182563408400, 62230116888000

Offset: 0

Alois P. Heinz, Jul 15 2016

nonn

			a(9) = 40: 159|26|37|48, 15|26|37|48|9, 159|26|37|4|8, 15|26|37|4|8|9, 159|26|3|48|7, 15|26|3|48|7|9, 159|26|3|4|7|8, 15|26|3|4|7|8|9, 159|2|37|48|6, 15|2|37|48|6|9, 159|2|37|4|6|8, 15|2|37|4|6|8|9, 159|2|3|48|6|7, 15|2|3|48|6|7|9, 159|2|3|4|6|7|8, 15|2|3|4|6|7|8|9, 19|26|37|48|5, 1|26|37|48|59, 1|26|37|48|5|9, 19|26|37|4|5|8, 1|26|37|4|59|8, 1|26|37|4|5|8|9, 19|26|3|48|5|7, 1|26|3|48|59|7, 1|26|3|48|5|7|9, 19|26|3|4|5|7|8, 1|26|3|4|59|7|8, 1|26|3|4|5|7|8|9, 19|2|37|48|5|6, 1|2|37|48|59|6, 1|2|37|48|5|6|9, 19|2|37|4|5|6|8, 1|2|37|4|59|6|8, 1|2|37|4|5|6|8|9, 19|2|3|48|5|6|7, 1|2|3|48|59|6|7, 1|2|3|48|5|6|7|9, 19|2|3|4|5|6|7|8, 1|2|3|4|59|6|7|8, 1|2|3|4|5|6|7|8|9.

Alois P. Heinz, Table of n, a(n) for n = 0..739
Wikipedia, Partition of a set

Column k=4 of A275069.

Cf. A000110.

a(n) = Product_{i=0..3} A000110(floor((n+i)/4)).

A275072 Number of set partitions of [n] such that i-j is a multiple of five for all i,j belonging to the same block.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 4, 8, 16, 32, 80, 200, 500, 1250, 3125, 9375, 28125, 84375, 253125, 759375, 2632500, 9126000, 31636800, 109674240, 380204032, 1484258048, 5794315072, 22620114608, 88305447412, 344730881243, 1489305334237, 6434092503083, 27796547414797

Offset: 0

Alois P. Heinz, Jul 15 2016

nonn

			a(9) = 16: 16|27|38|49|5, 16|27|38|4|5|9, 16|27|3|49|5|8, 16|27|3|4|5|8|9, 16|2|38|49|5|7, 16|2|38|4|5|7|9, 16|2|3|49|5|7|8, 16|2|3|4|5|7|8|9, 1|27|38|49|5|6, 1|27|38|4|5|6|9, 1|27|3|49|5|6|8, 1|27|3|4|5|6|8|9, 1|2|38|49|5|6|7, 1|2|38|4|5|6|7|9, 1|2|3|49|5|6|7|8, 1|2|3|4|5|6|7|8|9.

Alois P. Heinz, Table of n, a(n) for n = 0..772
Wikipedia, Partition of a set

Column k=5 of A275069.

Cf. A000110.

a(n) = Product_{i=0..4} A000110(floor((n+i)/5)).

A275073 Number of set partitions of [n] such that i-j is a multiple of six for all i,j belonging to the same block.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 4, 8, 16, 32, 64, 160, 400, 1000, 2500, 6250, 15625, 46875, 140625, 421875, 1265625, 3796875, 11390625, 39487500, 136890000, 474552000, 1645113600, 5703060480, 19770609664, 77181418496, 301304383744, 1176245959616, 4591883265424

Offset: 0

Alois P. Heinz, Jul 15 2016

nonn

			a(9) = 8: 17|28|39|4|5|6, 17|28|3|4|5|6|9, 17|2|39|4|5|6|8, 17|2|3|4|5|6|8|9, 1|28|39|4|5|6|7, 1|28|3|4|5|6|7|9, 1|2|39|4|5|6|7|8, 1|2|3|4|5|6|7|8|9.

Alois P. Heinz, Table of n, a(n) for n = 0..801
Wikipedia, Partition of a set

Column k=6 of A275069.

Cf. A000110.

a(n) = Product_{i=0..5} A000110(floor((n+i)/6)).

A275074 Number of set partitions of [n] such that i-j is a multiple of seven for all i,j belonging to the same block.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 4, 8, 16, 32, 64, 128, 320, 800, 2000, 5000, 12500, 31250, 78125, 234375, 703125, 2109375, 6328125, 18984375, 56953125, 170859375, 592312500, 2053350000, 7118280000, 24676704000, 85545907200, 296559144960, 1028071702528

Offset: 0

Alois P. Heinz, Jul 15 2016

nonn

			a(9) = 4: 18|29|3|4|5|6|7, 18|2|3|4|5|6|7|9, 1|29|3|4|5|6|7|8, 1|2|3|4|5|6|7|8|9.

Alois P. Heinz, Table of n, a(n) for n = 0..826
Wikipedia, Partition of a set

Column k=7 of A275069.

Cf. A000110.

a(n) = Product_{i=0..6} A000110(floor((n+i)/7)).

A275075 Number of set partitions of [n] such that i-j is a multiple of eight for all i,j belonging to the same block.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 640, 1600, 4000, 10000, 25000, 62500, 156250, 390625, 1171875, 3515625, 10546875, 31640625, 94921875, 284765625, 854296875, 2562890625, 8884687500, 30800250000, 106774200000, 370150560000, 1283188608000

Offset: 0

Alois P. Heinz, Jul 15 2016

nonn

			a(9) = 2: 19|2|3|4|5|6|7|8, 1|2|3|4|5|6|7|8|9.

Alois P. Heinz, Table of n, a(n) for n = 0..850
Wikipedia, Partition of a set

Column k=8 of A275069.

Cf. A000110.

a(n) = Product_{i=0..7} A000110(floor((n+i)/8)).

A275076 Number of set partitions of [n] such that i-j is a multiple of nine for all i,j belonging to the same block.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1280, 3200, 8000, 20000, 50000, 125000, 312500, 781250, 1953125, 5859375, 17578125, 52734375, 158203125, 474609375, 1423828125, 4271484375, 12814453125, 38443359375, 133270312500, 462003750000

Offset: 0

Alois P. Heinz, Jul 15 2016

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..871
Wikipedia, Partition of a set

Column k=9 of A275069.

Cf. A000110.

a(n) = Product_{i=0..8} A000110(floor((n+i)/9)).

A275077 Number of set partitions of [n] such that i-j is a multiple of ten for all i,j belonging to the same block.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2560, 6400, 16000, 40000, 100000, 250000, 625000, 1562500, 3906250, 9765625, 29296875, 87890625, 263671875, 791015625, 2373046875, 7119140625, 21357421875, 64072265625, 192216796875

Offset: 0

Alois P. Heinz, Jul 15 2016

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..891
Wikipedia, Partition of a set

Column k=10 of A275069.

Cf. A000110.

a(n) = Product_{i=0..9} A000110(floor((n+i)/10)).

cp's OEIS Frontend

A124419 Number of partitions of the set {1,2,...n} having no blocks that contain both odd and even entries.

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A275070 Number of set partitions of [n] such that i-j is a multiple of three for all i,j belonging to the same block.

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A275071 Number of set partitions of [n] such that i-j is a multiple of four for all i,j belonging to the same block.

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A275072 Number of set partitions of [n] such that i-j is a multiple of five for all i,j belonging to the same block.

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A275073 Number of set partitions of [n] such that i-j is a multiple of six for all i,j belonging to the same block.

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A275074 Number of set partitions of [n] such that i-j is a multiple of seven for all i,j belonging to the same block.

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A275075 Number of set partitions of [n] such that i-j is a multiple of eight for all i,j belonging to the same block.

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A275076 Number of set partitions of [n] such that i-j is a multiple of nine for all i,j belonging to the same block.

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A275077 Number of set partitions of [n] such that i-j is a multiple of ten for all i,j belonging to the same block.

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