A152875 -id:A152875 - 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

A152874 Triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} with k parity changes (n>=2; 1<=k <=n-1); the permutation 372185946 has 5 parity changes: 37-2-1-8-59-46.

Original entry on oeis.org

2, 4, 2, 8, 8, 8, 24, 36, 48, 12, 72, 144, 288, 144, 72, 288, 720, 1728, 1296, 864, 144, 1152, 3456, 10368, 10368, 10368, 3456, 1152, 5760, 20160, 69120, 86400, 103680, 51840, 23040, 2880, 28800, 115200, 460800, 691200, 1036800, 691200, 460800, 115200, 28800

Offset: 2

Emeric Deutsch, Dec 15 2008

Sum of entries in row n is n! (A000142(n)).
T(n,n-1) = A092186(n).
T(n,1) = A152875(n).
Sum_{k=1..n-1} k*T(n,k) = 2*A077613(n).

			T(4,3) = 8 because we have 1243, 1423, 4132, 4312, 2134, 2314, 3241 and 3421.
Triangle starts:
   2;
   4,   2;
   8,   8,   8;
  24,  36,  48,  12;
  72, 144, 288, 144,  72;
  ...

Alois P. Heinz, Rows n = 2..150, flattened

Cf. A000142, A077613, A092186.

T(2n,n) gives A363180.

ae := proc (n, k) if `mod`(k, 2) = 0 then 2*factorial(n)^2*binomial(n-1, (1/2)*k-1)*binomial(n-1, (1/2)*k) else 2*factorial(n)^2*binomial(n-1, (1/2)*k-1/2)^2 end if end proc: ao := proc (n, k) if `mod`(k, 2) = 0 then factorial(n)*factorial(n+1)*(binomial(n, (1/2)*k)*binomial(n-1, (1/2)*k-1)+binomial(n, (1/2)*k-1)*binomial(n-1, (1/2)*k)) else 2*factorial(n)*factorial(n+1)*binomial(n, (1/2)*k-1/2)*binomial(n-1, (1/2)*k-1/2) end if end proc: T := proc (n, k) if `mod`(n, 2) = 0 then ae((1/2)*n, k) else ao((1/2)*n-1/2, k) end if end proc: for n from 2 to 10 do seq(T(n, k), k = 1 .. n-1) end do; # yields sequence in triangular form
# second Maple program:
b:= proc(x, y, t) option remember; `if`(x+y=0, 1, `if`(x>0,
      b(x-1, y, z)*x, 0)+`if`(y>0, expand(b(y-1, x, z)*y*t), 0))
    end:
T:= n-> (h-> (p-> seq(coeff(p, z, i), i=1..n-1))(b(h, n-h, 1)))(iquo(n, 2)):
seq(T(n), n=2..12);  # Alois P. Heinz, May 23 2023

b[x_, y_, t_] := b[x, y, t] = If[x + y == 0, 1, If[x > 0, b[x - 1, y, z]*x, 0] + If[y > 0, Expand[b[y - 1, x, z]*y*t], 0]];
T[n_] := Table[Coefficient[#, z, i], {i, 1, n-1}]&[b[#, n-#, 1]]&[ Quotient[n, 2]];
Table[T[n], {n, 2, 12}] // Flatten (* Jean-François Alcover, Aug 16 2023, after Alois P. Heinz *)

T(2n,k) = (n!)^2*a(n,k), where a(n,k) is the number of lattice paths from (0,0) to (n,n) with steps N=(0,1) and E=(1,0) and having k turns;
a(n,k) = 2*binomial(n-1,k/2-1)*binomial(n-1,k/2) if k even;
a(n,k) = 2*(binomial(n-1,(k-1)/2))^2 if k odd.
T(2n+1,k) = n!*(n+1)!*b(n,k), where b(n,k) is the number of lattice paths from (0,0) to (n,n+1) with steps N=(0,1) and E=(1,0) and having k turns;
b(n,k) = binomial(n,k/2)*binomial(n-1,k/2-1) + binomial(n,k/2-1)*binomial(n-1,k/2) = (binomial(n,k/2))^2*k(2n-k+1)/(n(2n-k+2)) if k even;
b(n,k) = 2*binomial(n,(k-1)/2)*binomial(n-1,(k-1)/2) if k odd.

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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