A124419 -id:A124419 - cp's OEIS frontend

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

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

A124424 Triangle read by rows: T(n,k) is the number of partitions of the set {1,2,...,n}, having exactly k blocks consisting of entries of the same parity (0<=k<=n).

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 1, 2, 1, 1, 3, 4, 5, 2, 1, 7, 14, 16, 10, 4, 1, 25, 48, 61, 42, 20, 6, 1, 79, 194, 250, 200, 106, 38, 9, 1, 339, 820, 1145, 958, 569, 230, 66, 12, 1, 1351, 3794, 5554, 5096, 3251, 1486, 486, 112, 16, 1, 6721, 18960, 29101, 28010, 19110, 9470, 3477, 930, 175, 20, 1

Offset: 0

Emeric Deutsch, Nov 01 2006

Row sums are the Bell numbers (A000110). T(n,0)=A124425(n).

			T(4,2) = 5 because we have 13|24, 14|2|3, 1|2|34, 1|23|4 and 12|3|4.
Triangle starts:
  1;
  0,  1;
  1,  0,  1;
  1,  2,  1,  1;
  3,  4,  5,  2, 1;
  7, 14, 16, 10, 4, 1;
  ...

Alois P. Heinz, Rows n = 0..140, flattened

Cf. A000110, A124418, A124419, A124420, A124421, A124422, A124423, A124425, A363434.

T(2n,n) gives A363435.

Q[0]:=1: for n from 1 to 11 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 11 do P[n]:=sort(subs({s=t,x=1},Q[n])) od: for n from 0 to 11 do seq(coeff(P[n],t,j),j=0..n) od; # yields sequence in triangular form
# second Maple program:
b:= proc(g, u) option remember;
      add(Stirling2(g, k)*Stirling2(u, k)*k!, k=0..min(g, u))
    end:
T:= proc(n, k) local g, u; g:= floor(n/2); u:= ceil(n/2);
      add(add(add(binomial(g, i)*Stirling2(i, h)*binomial(u, j)*
      Stirling2(j, k-h)*b(g-i, u-j), j=k-h..u), i=h..g), h=0..k)
    end:
seq(seq(T(n,k), k=0..n), n=0..12);  # Alois P. Heinz, Oct 24 2013

b[g_, u_] := b[g, u] = Sum[StirlingS2[g, k]*StirlingS2[u, k]*k!, {k, 0, Min[g, u]}] ; T[n_, k_] := Module[{g, u}, g = Floor[n/2]; u = Ceiling[n/2]; Sum[ Sum[ Sum[ Binomial[g, i]*StirlingS2[i, h]*Binomial[u, j]*StirlingS2[j, k-h]*b[g-i, u-j], {j, k-h, u}], {i, h, g}], {h, 0, k}]]; Table[Table[T[n, k], {k, 0, n}], {n, 0, 12}] // Flatten (* Jean-François Alcover, Feb 19 2015, after Alois P. Heinz *)

The generating polynomial of row n is P[n](t)=Q[n](t,t,1), 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.

Sum_{k=0..n} k * T(n,k) = A363434(n). - Alois P. Heinz, Jun 01 2023

A124526 Triangle, read by rows, where T(n,k) = A049020([n/2],k)*A049020([(n+1)/2],k).

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 4, 9, 1, 10, 30, 6, 25, 100, 36, 1, 75, 370, 186, 10, 225, 1369, 961, 100, 1, 780, 5587, 4960, 750, 15, 2704, 22801, 25600, 5625, 225, 1, 10556, 101774, 136960, 39000, 2325, 21, 41209, 454276, 732736, 270400, 24025, 441, 1, 178031, 2199262, 4110512, 1849120, 217000, 6027, 28, 769129, 10647169, 23059204, 12645136, 1960000, 82369, 784, 1, 3630780, 55493841, 136074274, 87570056, 16787400, 944230, 13720, 36

Offset: 0

Paul D. Hanna, Nov 08 2006

Row n has 1+floor(n/2) terms.
T(n,0) = A124419(n).
A124418(n,k) = k!*T(n,k) (conjecture).
A000110(n) = Sum_{k=0..[n/2]} k!*T(n,k), where A000110 is the Bell numbers.
Inspired by triangle A124418 and the work of Emeric Deutsch.

			Triangle begins:
1;
1;
1, 1;
2, 3;
4, 9, 1;
10, 30, 6;
25, 100, 36, 1;
75, 370, 186, 10;
225, 1369, 961, 100, 1;
780, 5587, 4960, 750, 15;
2704, 22801, 25600, 5625, 225, 1;
10556, 101774, 136960, 39000, 2325, 21;
41209, 454276, 732736, 270400, 24025, 441, 1;
178031, 2199262, 4110512, 1849120, 217000, 6027, 28;
769129, 10647169, 23059204, 12645136, 1960000, 82369, 784, 1;
3630780, 55493841, 136074274, 87570056, 16787400, 944230, 13720, 36; ...

Cf. A124418, A124419, A049020; A124527, A124528, A124529.

S[n_, k_] = Sum[StirlingS2[n, i] Binomial[i, k], {i, 0, n}];
T[n_, k_] := S[Floor[n/2], k] S[Floor[(n+1)/2], k];
Table[T[n, k], {n, 0, 15}, {k, 0, Floor[n/2]}] // Flatten (* Jean-François Alcover, Nov 02 2020 *)

{T(n,k) = (n\2)!*((n+1)\2)!*polcoeff(polcoeff(exp((1+y)*(exp(x+x*O(x^n))-1)),n\2),k) *polcoeff(polcoeff(exp((1+y)*(exp(x+x*O(x^n))-1)),(n+1)\2),k)}
for(n=0,15, for(k=0,n\2, print1(T(n,k),", "));print(""))

T(n,k) = A049020([n/2],k) * A049020([(n+1)/2],k), where A049020(n,k) = Sum_{i=0..n} S2(n,i) * C(i,k) and S2(n,k) = (1/k!)*Sum_{j=0..k} (-1)^(k-j)*C(k,j)*j^n (the Stirling numbers of 2nd kind).

A152875 Number of permutations of {1,2,...,n} with all odd entries preceding all even entries or all even entries preceding all odd entries.

Original entry on oeis.org

1, 1, 2, 4, 8, 24, 72, 288, 1152, 5760, 28800, 172800, 1036800, 7257600, 50803200, 406425600, 3251404800, 29262643200, 263363788800, 2633637888000, 26336378880000, 289700167680000, 3186701844480000, 38240422133760000, 458885065605120000, 5965505852866560000

Offset: 0

Emeric Deutsch, Dec 15 2008

a(n) = A152874(n,1).

			a(4)=8 because we have 1324, 1342, 3124, 3142, 2413, 2431, 4213 and 4231.

Alois P. Heinz, Table of n, a(n) for n = 0..506

Cf. A124419, A152874.

a := proc (n) if `mod`(n, 2) = 0 then 2*factorial((1/2)*n)^2 else 2*factorial((1/2)*n-1/2)*factorial((1/2)*n+1/2) end if end proc: seq(a(n), n = 2 .. 25);
# second Maple program:
a:= n-> (h-> 2^signum(h)*h!*(n-h)!)(iquo(n, 2)):
seq(a(n), n=0..27);  # Alois P. Heinz, May 23 2023
# third Maple program:
a:= proc(n) option remember; `if`(n<4, n*(n-1)/2+1,
       n*(n-1)*a(n-2)/4 +a(n-1)/2)
    end:
seq(a(n), n=0..27);  # Alois P. Heinz, May 23 2023

a[n_] := Which[n<2, 1, EvenQ[n], 2(n/2)!^2, True, 2((n-1)/2)!*((n+1)/2)!];
Table[a[n], {n, 0, 27}] (* Jean-François Alcover, Aug 16 2023 *)

a(2n) = 2n!^2; a(2n+1) = 2n!(n+1)! (for n>=2).
E.g.f.: 1+x+2*(4*sqrt(4-x^2)*arcsin(x/2) - 4x + 4x^2 + x^3 - x^4)/((2+x)*(2-x)^2).
D-finite with recurrence 4*a(n) -2*a(n-1) -n*(n-1)*a(n-2)=0. - R. J. Mathar, Jul 22 2022

a(0)=a(1)=1 prepended by Alois P. Heinz, May 23 2023

A363454 Number of partitions of [n] such that the number of blocks containing only odd elements equals the number of blocks containing only even elements and no block contains both odd and even elements.

Original entry on oeis.org

1, 0, 1, 1, 2, 4, 11, 28, 87, 266, 952, 3381, 13513, 53915, 237113, 1046732, 5016728, 24186664, 125121009, 652084528, 3615047527, 20211789423, 119384499720, 711572380960, 4455637803543, 28162688795697, 186152008588691, 1242276416218540, 8636436319397292

Offset: 0

Alois P. Heinz, Jun 02 2023

nonn

			a(0) = 1: () the empty partition.
a(1) = 0.
a(2) = 1: 1|2.
a(3) = 1: 13|2.
a(4) = 2: 13|24, 1|2|3|4.
a(5) = 4: 135|24, 13|2|4|5, 15|2|3|4, 1|2|35|4.
a(6) = 11: 135|246, 13|24|5|6, 13|26|4|5, 13|2|46|5, 15|24|3|6, 1|24|35|6, 15|26|3|4, 15|2|3|46, 1|26|35|4, 1|2|35|46, 1|2|3|4|5|6.
a(7) = 28: 1357|246, 135|24|6|7, 137|24|5|6, 13|24|57|6, 135|26|4|7, 135|2|46|7, 137|26|4|5, 13|26|4|57, 137|2|46|5, 13|2|46|57, 13|2|4|5|6|7, 157|24|3|6, 15|24|37|6, 17|24|35|6, 1|24|357|6, 157|26|3|4, 15|26|37|4, 157|2|3|46, 15|2|37|46, 15|2|3|4|6|7, 17|26|35|4, 1|26|357|4, 17|2|35|46, 1|2|357|46, 1|2|35|4|6|7, 17|2|3|4|5|6, 1|2|37|4|5|6, 1|2|3|4|57|6.

Alois P. Heinz, Table of n, a(n) for n = 0..650
Wikipedia, Partition of a set

Bisection gives A047797 (even part).

Cf. A000110, A124419, A124425, A363451.

a:= n-> (h-> add(Stirling2(h, k)*Stirling2(n-h, k), k=0..h))(iquo(n, 2)):
seq(a(n), n=0..40);
# second Maple program:
b:= proc(n, x, y) option remember; `if`(abs(x-y)>n, 0,
      `if`(n=0, 1, `if`(x>0, b(n-1, y, x)*x, 0)+b(n-1, y, x+1)))
    end:
a:= n-> b(n, 0$2):
seq(a(n), n=0..40);

a(n) = Sum_{k=0..floor(n/2)} Stirling2(floor(n/2),k) * Stirling2(ceiling(n/2),k).

a(2n) = A047797(n).

A363511 Number of partitions of [n] having exactly one parity change within their blocks.

Original entry on oeis.org

0, 0, 1, 2, 6, 18, 61, 210, 778, 3008, 12219, 52268, 231726, 1083012, 5202199, 26307710, 135972580, 738339310, 4081523615, 23649300862, 139096468520, 855529383396, 5329630673249, 34643027568520, 227682351175868, 1558106351450416, 10766192988109009

Offset: 0

Alois P. Heinz, Jun 06 2023

nonn

			a(4) = 6: 124|3, 12|3|4, 134|2, 1|23|4, 14|2|3, 1|2|34.

Alois P. Heinz, Table of n, a(n) for n = 0..500
Wikipedia, Partition of a set

Column k=1 of A363493.

Cf. A000110, A124419.

b:= proc(n, x, y) option remember; convert(series(
     `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)), z, 2), polynom)
    end:
a:= n-> coeff(b(n, 0$2), z, 1):
seq(a(n), n=0..27);

A124426 Product of two successive Bell numbers.

Original entry on oeis.org

1, 2, 10, 75, 780, 10556, 178031, 3630780, 87548580, 2452523325, 78697155750, 2859220516290, 116482516809889, 5277304280371714, 264005848594606490, 14493602135008296115, 868435614538568029188, 56520205738693680322836

Offset: 0

Emeric Deutsch and Paul D. Hanna, Nov 03 2006

nonn

Number of partitions of the set {1,2,...,2n+1} having no blocks that contain both odd and even entries. Example: a(2)=10 because we have 135|24, 15|24|3, 1|24|35, 135|2|4, 15|2|3|4, 1|2|35|4, 13|24|5, 1|24|3|5, 13|2|4|5 and 1|2|3|4|5. a(n)=A124419(2n+1)=A124418(2n+1,0).

Harvey P. Dale, Table of n, a(n) for n = 0..324

Cf. A000110, A008277, A124418, A124419.

[&*[ Bell(n+k): k in [0..1] ]: n in [0..30]]; // Vincenzo Librandi, Apr 09 2020

with(combinat): seq(bell(n)*bell(n+1),n=0..19);

Times@@@Partition[BellB[Range[0,20]],2,1] (* Harvey P. Dale, Oct 07 2018 *)

a(n) = B(n)*B(n+1), where B(q) are the Bell numbers (A000110), i.e., B(n) = Sum_{k=1..n} S2(n,k), S2(n,k) being the Stirling numbers of the 2nd kind (A008277).

A363073 Number of set partitions of [n] such that each element is contained in a block whose block size parity coincides with the parity of the element.

Original entry on oeis.org

1, 1, 0, 0, 1, 2, 0, 0, 20, 48, 0, 0, 1147, 3968, 0, 0, 173203, 709488, 0, 0, 53555964, 246505600, 0, 0, 28368601065, 148963383616, 0, 0, 24044155851601, 141410718244864, 0, 0, 30934515698084780, 198914201874983936, 0, 0, 57215369885233295955, 398742900995358584320

Offset: 0

Alois P. Heinz, May 17 2023

nonn

All odd elements are in blocks with an odd block size and all even elements are in blocks with an even block size.

			a(0) = 1: (), the empty partition.
a(1) = 1: 1.
a(4) = 1: 1|24|3.
a(5) = 2: 135|24, 1|24|3|5.
a(8) = 20: 135|2468|7, 135|24|68|7, 137|2468|5, 137|24|5|68, 135|26|48|7, 135|28|46|7, 137|26|48|5, 137|28|46|5, 157|2468|3, 157|24|3|68, 1|2468|357, 1|24|357|68, 1|2468|3|5|7, 1|24|3|5|68|7, 157|26|3|48, 157|28|3|46, 1|26|357|48, 1|28|357|46, 1|26|3|48|5|7, 1|28|3|46|5|7.

Alois P. Heinz, Table of n, a(n) for n = 0..672
Wikipedia, Partition of a set

Cf. A000110, A003724, A005046, A124419, A274538, A275679.

b:= proc(n, t) option remember; `if`(n=0, 1, add(
     `if`((j+t)::even, b(n-j, t)*binomial(n-1, j-1), 0), j=1..n))
    end:
a:= n-> (h-> b(n-h, 1)*b(h, 0))(iquo(n, 2)):
seq(a(n), n=0..40);

b[n_, t_] := b[n, t] = If[n == 0, 1, Sum[If[EvenQ[j + t], b[n - j, t]* Binomial[n - 1, j - 1], 0], {j, 1, n}]];
a[n_] := b[n - #, 1]*b[#, 0]&[Quotient[n, 2]];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Nov 18 2023, after Alois P. Heinz *)

a(n) = A003724(ceiling(n/2)) * A005046(floor(n/4)) if (n mod 4) in {0,1}.

a(n) = 0 if (n mod 4) in {2,3}.

cp's OEIS Frontend

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

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A124424 Triangle read by rows: T(n,k) is the number of partitions of the set {1,2,...,n}, having exactly k blocks consisting of entries of the same parity (0<=k<=n).

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A124526 Triangle, read by rows, where T(n,k) = A049020([n/2],k)*A049020([(n+1)/2],k).

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A152875 Number of permutations of {1,2,...,n} with all odd entries preceding all even entries or all even entries preceding all odd entries.

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A363454 Number of partitions of [n] such that the number of blocks containing only odd elements equals the number of blocks containing only even elements and no block contains both odd and even elements.

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A363511 Number of partitions of [n] having exactly one parity change within their blocks.

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A124426 Product of two successive Bell numbers.

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A363073 Number of set partitions of [n] such that each element is contained in a block whose block size parity coincides with the parity of the element.