A274547
Number of set partitions of [n] with alternating parity of elements.
Original entry on oeis.org
1, 1, 2, 4, 8, 18, 40, 101, 254, 723, 2064, 6586, 21143, 74752, 266078, 1029983, 4013425, 16843526, 71136112, 321150717, 1458636308, 7038678613, 34161890155, 175261038904, 904125989974, 4909033438008, 26795600521492, 153376337926066, 882391616100249
Offset: 0
a(5) = 18: 12345, 1234|5, 123|45, 123|4|5, 12|345, 12|34|5, 12|3|45, 12|3|4|5, 145|23, 1|2345, 1|234|5, 1|23|45, 1|23|4|5, 145|2|3, 1|2|345, 1|2|34|5, 1|2|3|45, 1|2|3|4|5.
a(6) = 40: 123456, 12345|6, 1234|56, 1234|5|6, 123|456, 123|45|6, 123|4|56, 123|4|5|6, 1256|34, 12|3456, 12|345|6, 12|34|56, 12|34|5|6, 1256|3|4, 12|3|456, 12|3|45|6, 12|3|4|56, 12|3|4|5|6, 145|236, 145|23|6, 1|23456, 1|2345|6, 1|234|56, 1|234|5|6, 1|23|456, 1|23|45|6, 1|23|4|56, 1|23|4|5|6, 145|2|36, 145|2|3|6, 1|256|34, 1|2|3456, 1|2|345|6, 1|2|34|56, 1|2|34|5|6, 1|256|3|4, 1|2|3|456, 1|2|3|45|6, 1|2|3|4|56, 1|2|3|4|5|6.
-
b:= proc(l, i, t) option remember; `if`(l=[], 1, add(`if`(l[j]=t,
b(subsop(j=[][], l), j, 1-t), 0), j=[1, $i..nops(l)]))
end:
a:= n-> b([seq(irem(i, 2), i=2..n)], 1, 0):
seq(a(n), n=0..25);
-
b[l_, i_, t_] := b[l, i, t] = If[l == {}, 1, Sum[If[l[[j]] == t, b[ReplacePart[l, j -> Sequence[]], j, 1-t], 0], {j, Prepend[Range[i, Length[l]], 1]}]]; a[n_] := b[Table[Mod[i, 2], {i, 2, n}], 1, 0]; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Feb 15 2017, translated from Maple *)
A274860
Number of set partitions of [n] such that the difference between each element and its index (in the partition) is a multiple of three.
Original entry on oeis.org
1, 1, 2, 4, 8, 16, 32, 68, 144, 304, 692, 1580, 3602, 8976, 22552, 56850, 157740, 442101, 1249124, 3868420, 12082568, 38117838, 130394944, 448733361, 1559590892, 5817284716, 21787658638, 82365982741, 331668252301, 1339477442569, 5457855667219, 23555837192320
Offset: 0
a(7) = 68: 1234567, 123456|7, 12345|67, 12345|6|7, 1234|567, 1234|56|7, 1234|5|67, 1234|5|6|7, 123|4567, 123|456|7, 123|45|67, 123|45|6|7, 123|4|567, 123|4|56|7, 123|4|5|67, 123|4|5|6|7, 12|34567, 12|3456|7, 12|345|67, 12|345|6|7, 12|34|567, 12|34|56|7, 12|34|5|67, 12|34|5|6|7, 12|3|4567, 12|3|456|7, 12|3|45|67, 12|3|45|6|7, 12|3|4|567, 12|3|4|56|7, 12|3|4|5|67, 12|3|4|5|6|7, 1567|234, 1|234567, 1|23456|7, 1|2345|67, 1|2345|6|7, 1|234|567, 1|234|56|7, 1|234|5|67, 1|234|5|6|7, 1567|23|4, 1|23|4567, 1|23|456|7, 1|23|45|67, 1|23|45|6|7, 1|23|4|567, 1|23|4|56|7, 1|23|4|5|67, 1|23|4|5|6|7, 1567|2|34, 1|2|34567, 1|2|3456|7, 1|2|345|67, 1|2|345|6|7, 1|2|34|567, 1|2|34|56|7, 1|2|34|5|67, 1|2|34|5|6|7, 1567|2|3|4, 1|2|3|4567, 1|2|3|456|7, 1|2|3|45|67, 1|2|3|45|6|7, 1|2|3|4|567, 1|2|3|4|56|7, 1|2|3|4|5|67, 1|2|3|4|5|6|7.
A274861
Number of set partitions of [n] such that the difference between each element and its index (in the partition) is a multiple of four.
Original entry on oeis.org
1, 1, 2, 4, 8, 16, 32, 64, 128, 264, 544, 1120, 2304, 4940, 10632, 22908, 49324, 112352, 258184, 596430, 1379150, 3431265, 8639020, 21965049, 56062850, 155582125, 436328469, 1238168841, 3534962983, 10951026755, 34135076965, 107562008116, 341226376721
Offset: 0
A274862
Number of set partitions of [n] such that the difference between each element and its index (in the partition) is a multiple of five.
Original entry on oeis.org
1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1040, 2112, 4288, 8704, 17664, 36640, 76208, 158712, 330632, 688600, 1481712, 3210320, 6987992, 15244544, 33265812, 76267252, 176853756, 413956038, 974437212, 2297088912, 5798212320, 14820017916, 38348190276
Offset: 0
A274863
Number of set partitions of [n] such that the difference between each element and its index (in the partition) is a multiple of six.
Original entry on oeis.org
1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4128, 8320, 16768, 33792, 68096, 137216, 279632, 570784, 1166128, 2383312, 4871312, 9955760, 20713120, 43286304, 90751688, 190621800, 400733360, 842536776, 1825618200, 3990868296, 8789613840, 19459895108
Offset: 0
A274864
Number of set partitions of [n] such that the difference between each element and its index (in the partition) is a multiple of seven.
Original entry on oeis.org
1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16448, 33024, 66304, 133120, 267264, 536576, 1077248, 2175168, 4395968, 8888800, 17978016, 36364960, 73558304, 148788576, 303793408, 621878080, 1275478720, 2619224480, 5382273968, 11063449024
Offset: 0
A274865
Number of set partitions of [n] such that the difference between each element and its index (in the partition) is a multiple of eight.
Original entry on oeis.org
1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65664, 131584, 263680, 528384, 1058816, 2121728, 4251648, 8519680, 17121728, 34424960, 69234368, 139262272, 280139584, 563542336, 1133653568, 2280506048, 4609715712, 9330877056
Offset: 0
A274866
Number of set partitions of [n] such that the difference between each element and its index (in the partition) is a multiple of nine.
Original entry on oeis.org
1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262400, 525312, 1051648, 2105344, 4214784, 8437760, 16891904, 33816576, 67698688, 135726080, 272175872, 545883008, 1094922880, 2196262528, 4405472896, 8836975232
Offset: 0
A274867
Number of set partitions of [n] such that the difference between each element and its index (in the partition) is a multiple of ten.
Original entry on oeis.org
1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144, 524288, 1049088, 2099200, 4200448, 8404992, 16818176, 33652736, 67338240, 134742016, 269615104, 539492352, 1080297728, 2163485184, 4333081344, 8678753536
Offset: 0
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