A274581 -id:A274581 - cp's OEIS frontend

A274310 Triangle read by rows: T(n,k) = number of parity alternating partitions of [n] into k blocks (1 <= k <= m).

1, 1, 1, 1, 2, 1, 1, 4, 4, 1, 1, 6, 11, 6, 1, 1, 10, 28, 26, 9, 1, 1, 14, 61, 86, 50, 12, 1, 1, 22, 136, 276, 236, 92, 16, 1, 1, 30, 275, 770, 927, 530, 150, 20, 1, 1, 46, 580, 2200, 3551, 2782, 1130, 240, 25, 1, 1, 62, 1141, 5710, 12160, 12632, 6987, 2130, 355, 30, 1

Offset: 1

N. J. A. Sloane, Jun 23 2016

The first element of any block may be odd or even and then the parity of terms alternates within each block. - Alois P. Heinz, Jun 28 2016

Let a(n,k,i) be the number of parity alternating partitions of n into k blocks, i of which have even maximal elements. Dzhumadil'daev and Yeliussizov, Proposition 5.3, give recurrences for a(n,k,i), which depend on the parity of n. It is easy to verify that the solution to these recurrences is given by a(2*n,k,i) = Stirling2(n,i)*Stirling2(n+1,k+1-i) and a(2*n+1,k,i) = Stirling2(n+1,i+1) * Stirling2(n+1,k-i). The formula below for the table entries T(n,k) follows from this observation. - Peter Bala, Apr 09 2018

			Triangle begins:
  1;
  1,   1;
  1,   2,   1;
  1,   4,   4,   1;
  1,   6,  11,   6,   1;
  1,  10,  28,  26,   9,   1;
  1,  14,  61,  86,  50,  12,   1;
  1,  22, 136, 276, 236,  92,  16,   1;
  ...
From _Alois P. Heinz_, Jun 28 2016: (Start)
T(5,1) = 1: 12345.
T(5,2) = 6: 1234|5, 123|45, 125|34, 12|345, 145|23, 1|2345.
T(5,3) = 11: 123|4|5, 12|34|5, 125|3|4, 12|3|45, 14|23|5, 1|234|5, 1|23|45, 145|2|3, 14|25|3, 1|25|34, 1|2|345.
T(5,4) = 6: 12|3|4|5, 1|23|4|5, 14|2|3|5, 1|2|34|5, 1|25|3|4, 1|2|3|45.
T(5,5) = 1: 1|2|3|4|5. (End)

Alois P. Heinz, Rows n = 1..141, flattened
Askar Dzhumadil'daev and Damir Yeliussizov, Walks, partitions, and normal ordering, Electronic Journal of Combinatorics, 22(4) (2015), #P4.10.

Row sums give A124419(n+1).

Cf. A274547, A274581.

A274310 := proc (n, k) local i;
with(combinat):
   add(Stirling2(floor((1/2)*n+1), i+1)*Stirling2(floor((1/2)*n+1/2), k-i), i = 0..k-1);
end proc:
for n from 1 to 10 do
   seq(A274310(n, k), k = 1..n);
end do; # Peter Bala, Apr 09 2018

T[n_, k_] = Sum[StirlingS2[Floor[(n + 2)/2], i + 1] * StirlingS2[Floor[(n + 1)/2], k - i], {i, 0, k - 1}];
Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Jean-François Alcover, May 17 2018, after Peter Bala *)

T(n,k) = Sum_{i = 0..k-1} Stirling2(floor((n+2)/2), i+1) * Stirling2(floor((n+1)/2), k-i). - Peter Bala, Apr 09 2018

More terms from Alois P. Heinz, Jun 26 2016

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

Alois P. Heinz, Jun 27 2016

nonn

			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.

Alois P. Heinz, Table of n, a(n) for n = 0..36
Wikipedia, Partition of a set

Row sums of A274581.
Cf. A124419, A274310 (parities alternate within blocks), A363519.
Column k=2 of A274859.

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 *)

a(n) = Sum_{k=0..n} A274581(n,k).

a(n) = A363519(n,max(0,n-1)). - Alois P. Heinz, Jun 07 2023

A305777 Number of set partitions of [n] with alternating parity of elements and exactly three blocks.

Original entry on oeis.org

1, 3, 7, 14, 30, 57, 119, 224, 460, 867, 1761, 3338, 6734, 12861, 25843, 49748, 99744, 193431, 387365, 756062, 1513138, 2969265, 5940567, 11708072, 23420228, 46317195, 92642569, 183707186, 367430742, 730111269, 1460255291, 2906227196, 5812519912, 11582124159

Offset: 3

Alois P. Heinz, Jun 10 2018

			a(5) = 7: 123|4|5, 12|34|5, 12|3|45, 1|234|5, 145|2|3, 1|2|345, 1|23|45.

Alois P. Heinz, Table of n, a(n) for n = 3..1000
Wikipedia, Partition of a set
Index entries for linear recurrences with constant coefficients, signature (3,4,-18,1,33,-16,-18,12)

Column k=3 of A274581.

G.f.: (10*x^6 -x^5 -13*x^4 +x^3 +6*x^2-1) *x^3 / ((2*x-1) *(x+1) *(3*x^2-1) *(2*x^2-1) *(x-1)^2).

A305778 Number of set partitions of [n] with alternating parity of elements and exactly four blocks.

Original entry on oeis.org

1, 4, 12, 33, 84, 222, 545, 1425, 3458, 8948, 21595, 55335, 133144, 338530, 813249, 2056245, 4935870, 12432432, 29833463, 74952867, 179842724, 451071998, 1082309293, 2711584785, 6506473162, 16289249356, 39088013091, 97811687679, 234720639024, 587166227226

Offset: 4

Alois P. Heinz, Jun 10 2018

			a(5) = 4: 12|3|4|5, 1|23|4|5, 1|2|34|5, 1|2|3|45.

Alois P. Heinz, Table of n, a(n) for n = 4..1000
Wikipedia, Partition of a set
Index entries for linear recurrences with constant coefficients, signature (1,18,-18,-127,127,450,-450,-844,844,792,-792,-288,288)

Column k=4 of A274581.

G.f.: (72*x^11 -84*x^10 -258*x^9 +60*x^8 +287*x^7 +29*x^6 -141*x^5 -34*x^4 +33*x^3 +10*x^2 -3*x-1) *x^4 / ((2*x+1) *(2*x-1) *(x+1) *(6*x^2-1) *(3*x^2-1) *(x-1)^2 *(2*x^2-1)^2).

A305779 Number of set partitions of [n] with alternating parity of elements and exactly five blocks.

Original entry on oeis.org

1, 5, 19, 62, 204, 627, 2006, 6045, 19091, 56804, 177431, 523072, 1618006, 4739853, 14544372, 42440871, 129419597, 376877914, 1143885829, 3328765378, 10068808024, 29308488795, 88433869354, 257651383313, 776063514999, 2264100874960, 6811218232163, 19903182431380

Offset: 5

Alois P. Heinz, Jun 10 2018

nonn

Wikipedia, Partition of a set

Column k=5 of A274581.

A305780 Number of set partitions of [n] with alternating parity of elements and exactly six blocks.

Original entry on oeis.org

1, 6, 27, 108, 409, 1558, 5692, 21328, 76468, 282885, 999383, 3656887, 12772925, 46306978, 160408850, 577077226, 1987559778, 7105490315, 24379316121, 86719101209, 296819500851, 1051663900480, 3594472289644, 12696988924724, 43364647235536, 152825304753761

Offset: 6

Alois P. Heinz, Jun 10 2018

nonn

Wikipedia, Partition of a set

Column k=6 of A274581.

A305781 Number of set partitions of [n] with alternating parity of elements and exactly seven blocks.

Original entry on oeis.org

1, 7, 37, 169, 763, 3287, 14402, 60752, 261772, 1085823, 4614736, 18868613, 79286687, 320425251, 1333901662, 5341972670, 22066221462, 87764483463, 360206083844, 1425407602339, 5819322422329, 22944438320083, 93269292846714, 366809712490908, 1485913213777040

Offset: 7

Alois P. Heinz, Jun 10 2018

nonn

Wikipedia, Partition of a set

Column k=7 of A274581.

A305782 Number of set partitions of [n] with alternating parity of elements and exactly eight blocks.

Original entry on oeis.org

1, 8, 48, 254, 1284, 6456, 31559, 155828, 747046, 3633393, 17131855, 82235382, 382314833, 1814891650, 8339187411, 39220306016, 178502457056, 832998001071, 3762499165935, 17443657250690, 78322897989851, 361141330169264, 1614162421290491, 7409032399939348

Offset: 8

Alois P. Heinz, Jun 10 2018

nonn

Wikipedia, Partition of a set

Column k=8 of A274581.

A305783 Number of set partitions of [n] with alternating parity of elements and exactly nine blocks.

Original entry on oeis.org

1, 9, 61, 359, 2071, 11521, 64536, 351890, 1937292, 10385051, 56327364, 297361209, 1592259558, 8294350334, 43929021137, 226237017737, 1187116072881, 6055309005874, 31524381075907, 159529143743414, 825010507438945, 4148043681253075, 21331478060780614

Offset: 9

Alois P. Heinz, Jun 10 2018

nonn

Wikipedia, Partition of a set

Column k=9 of A274581.

A305784 Number of set partitions of [n] with alternating parity of elements and exactly ten blocks.

Original entry on oeis.org

1, 10, 75, 495, 3135, 19657, 120678, 743321, 4480240, 27176489, 161160411, 964119615, 5635200174, 33302382467, 192186538999, 1123752073702, 6413666231367, 37159362571856, 210072026510678, 1207564204033112, 6771732706003311, 38664907592673960, 215359228721631996

Offset: 10

Alois P. Heinz, Jun 10 2018

nonn

Wikipedia, Partition of a set

Column k=10 of A274581.

cp's OEIS Frontend

A274310 Triangle read by rows: T(n,k) = number of parity alternating partitions of [n] into k blocks (1 <= k <= m).

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A274547 Number of set partitions of [n] with alternating parity of elements.

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A305777 Number of set partitions of [n] with alternating parity of elements and exactly three blocks.

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A305778 Number of set partitions of [n] with alternating parity of elements and exactly four blocks.

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A305779 Number of set partitions of [n] with alternating parity of elements and exactly five blocks.

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A305780 Number of set partitions of [n] with alternating parity of elements and exactly six blocks.

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A305781 Number of set partitions of [n] with alternating parity of elements and exactly seven blocks.

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A305782 Number of set partitions of [n] with alternating parity of elements and exactly eight blocks.

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A305783 Number of set partitions of [n] with alternating parity of elements and exactly nine blocks.

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A305784 Number of set partitions of [n] with alternating parity of elements and exactly ten blocks.

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