A363493
Number T(n,k) of partitions of [n] having exactly k parity changes within their blocks, n>=0, 0<=k<=max(0,n-1), read by rows.
Original entry on oeis.org
1, 1, 1, 1, 2, 2, 1, 4, 6, 4, 1, 10, 18, 17, 6, 1, 25, 61, 68, 38, 10, 1, 75, 210, 292, 202, 83, 14, 1, 225, 778, 1252, 1116, 576, 170, 22, 1, 780, 3008, 5670, 5928, 3899, 1490, 341, 30, 1, 2704, 12219, 26114, 32382, 25320, 12655, 3856, 678, 46, 1, 10556, 52268, 126073, 177666, 163695, 98282, 39230, 9418, 1319, 62, 1
Offset: 0
T(4,0) = 4: 13|24, 13|2|4, 1|24|3, 1|2|3|4.
T(4,1) = 6: 124|3, 12|3|4, 134|2, 1|23|4, 14|2|3, 1|2|34.
T(4,2) = 4: 123|4, 12|34, 14|23, 1|234.
T(4,3) = 1: 1234.
T(5,2) = 17: 1235|4, 123|4|5, 1245|3, 12|34|5, 125|3|4, 12|3|45, 1345|2, 134|25, 14|235, 14|23|5, 15|234, 1|234|5, 1|23|45, 145|2|3, 14|25|3, 1|25|34, 1|2|345.
Triangle T(n,k) begins:
1;
1;
1, 1;
2, 2, 1;
4, 6, 4, 1;
10, 18, 17, 6, 1;
25, 61, 68, 38, 10, 1;
75, 210, 292, 202, 83, 14, 1;
225, 778, 1252, 1116, 576, 170, 22, 1;
780, 3008, 5670, 5928, 3899, 1490, 341, 30, 1;
2704, 12219, 26114, 32382, 25320, 12655, 3856, 678, 46, 1;
...
-
b:= proc(n, x, y) option remember; `if`(n=0, 1,
`if`(y=0, 0, expand(b(n-1, y-1, x+1)*y*z))+
b(n-1, y, x)*x + b(n-1, y, x+1))
end:
T:= n-> (p-> seq(coeff(p, z, i), i=0..degree(p)))(b(n, 0$2)):
seq(T(n), n=0..12);
-
b[n_, x_, y_] := b[n, x, y] = If[n == 0, 1,
If[y == 0, 0, Expand[b[n - 1, y - 1, x + 1]*y*z]] +
b[n - 1, y, x]*x + b[n - 1, y, x + 1]];
T[n_] := CoefficientList[b[n, 0, 0], z];
Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Sep 05 2023, after Alois P. Heinz *)
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 *)
A363549
Total number of parity changes within all partitions of [n].
Original entry on oeis.org
0, 0, 2, 9, 35, 152, 690, 3476, 18362, 105507, 633439, 4077386, 27317322, 194202164, 1431942154, 11117639073, 89255122863, 750051307896, 6500488706254, 58693723144324, 545325034579386, 5258317906940219, 52072378163037347, 533479904210984930, 5603461526474415150
Offset: 0
a(3) = 9 = 1*1 + 4*2: 13|2, 123, 12|3, 1|23, 1|2|3.
a(4) = 35 = 3*1 + 4*2 + 8*3: 134|2, 13|24, 13|2|4, 124|3, 14|23, 14|2|3, 1|24|3, 1234, 123|4, 12|34, 12|3|4, 1|234, 1|23|4, 1|2|34, 1|2|3|4.
A363550
Number of partitions of [n] having exactly one parity change within the partition.
Original entry on oeis.org
0, 0, 2, 1, 3, 2, 7, 5, 20, 15, 67, 52, 255, 203, 1080, 877, 5017, 4140, 25287, 21147, 137122, 115975, 794545, 678570, 4892167, 4213597, 31858034, 27644437, 218543759, 190899322, 1573857867, 1382958545, 11863100692, 10480142147, 93345011951, 82864869804
Offset: 0
a(2) = 2: 12, 1|2.
a(3) = 1: 13|2.
a(4) = 3: 134|2, 13|24, 13|2|4.
a(5) = 2: 135|24, 135|2|4.
a(6) = 7: 1356|24, 135|246, 135|24|6, 1356|2|4, 135|26|4, 135|2|46, 135|2|4|6.
a(7) = 5: 1357|246, 1357|24|6, 1357|26|4, 1357|2|46, 1357|2|4|6.
a(8) = 20: 13578|246, 1357|2468, 1357|246|8, 13578|24|6, 1357|248|6, 1357|24|68, 1357|24|6|8, 13578|26|4, 1357|268|4, 1357|26|48, 1357|26|4|8, 13578|2|46, 1357|28|46, 1357|2|468, 1357|2|46|8, 13578|2|4|6, 1357|28|4|6, 1357|2|48|6, 1357|2|4|68, 1357|2|4|6|8.
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b:= proc(n) option remember; `if`(n=0, 1,
add(b(n-j)*binomial(n-1, j-1), j=1..n))
end:
a:= n-> `if`(n<2, 0, (h-> b(h)+`if`(n::even, b(h-1), 0))(iquo(n, 2))):
seq(a(n), n=0..35);
Showing 1-4 of 4 results.
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