A060350 -id:A060350 - cp's OEIS frontend

A291902 Sums of the cubes of the descent set statistics for permutations on n elements.

1, 1, 2, 18, 360, 14460, 994680, 109021500, 17815754880, 4147063256448, 1323985303267200, 562636176102554400, 310405397451855552000, 217731000904433587359360, 190749857434239995742090240, 205540893695782384696324368000, 268793206446238988670401236992000

Offset: 0

Richard Ehrenborg, Sep 05 2017

nonn

			For n=4, we have a(4) = 1^3 + 3^3 + 5^5 + 3^3 + 3^3 + 5^3 + 3^3 + 1^3 = 360.

R. Ehrenborg and A. Happ, On the powers of the descent set statistic, arXiv:1709.00778 [math.CO], 2017.

Cf. A060350, A291903, A060351.

Column k=3 of A334622.

a(n) = Sum_{j=0..ceiling(2^(n-1))-1} A060351(n,j)^3. - Alois P. Heinz, Sep 15 2020

a(0)=1 prepended by Alois P. Heinz, Sep 09 2020

A291907 Numbers such that the nonzero digits in the base 3 expansion consists of two 1s and one 2.

Original entry on oeis.org

14, 16, 22, 32, 34, 38, 42, 46, 48, 58, 64, 66, 86, 88, 92, 96, 100, 102, 110, 114, 126, 136, 138, 144, 166, 172, 174, 190, 192, 198, 248, 250, 254, 258, 262, 264, 272, 276, 288, 298, 300, 306, 326, 330, 342, 378, 406, 408, 414, 432, 490, 496, 498, 514, 516, 522

Offset: 1

Richard Ehrenborg, Sep 05 2017

If k belongs to this sequence, A060350(k) and A291903(k) are divisible by 3.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Richard Ehrenborg and Alex Happ, On the powers of the descent set statistic, arXiv:1709.00778 [math.CO], 2017.

Cf. A007089, A023693, A023699, A060350, A291903.

Select[Range[1000], DigitCount[#, 3, {1, 2}] == {2, 1} &] (* Amiram Eldar, Apr 07 2022 *)

A023693 INTERSECT A023699. - R. J. Mathar, Nov 10 2017

A137783 a(n) = the number of permutations (p(1), p(2), ..., p(2n+1)) of (1, 2, ..., 2n+1) where, for each k (2 <= k <= 2n+1), the sign of (p(k) - p(k-1)) equals the sign of (p(2n+2-k) - p(2n+3-k)).

Original entry on oeis.org

1, 4, 44, 1028, 40864, 2484032, 214050784, 24831582176, 3731039384576, 704879630525696, 163539441616948736, 45712130697710081024, 15150993151215400441856, 5875388829103413298173952, 2635427286694074346846232576, 1353918066433734600362650169344

Offset: 0

Leroy Quet, Feb 10 2008, Feb 14 2008

nonn

There are no such permutations of (1,2,...,2n).

			Consider the permutation (for n = 3): 3,4,5,2,7,6,1. The signs of the differences between adjacent terms form the sequence: ++-+--, which is the negative of its reversal. So this permutation, among others, is counted when n = 3.

Alois P. Heinz, Table of n, a(n) for n = 0..90

Cf. A060350, A137782.

{ a(n) = my(s,c,r); s=0; forvec(t=vector(n\2,i,[0,2]), c=0; r=[]; for(j=1,#t, if(t[j]==0,c++, if(t[j]==1,r=concat(r,[j]),r=concat(r,[n-j])); ); ); r=vecsort(r); s+=(-2)^c*if(#r,n!/(r[1]!*prod(j=1,#r-1,(r[j+1]-r[j])!)*(n-r[ #r])!),1) ); s } /* Max Alekseyev */

First 4 terms calculated by Olivier Gérard

Edited and extended by Max Alekseyev, May 09 2009

A362745 Triangular array read by rows. T(n,k) is the number of ordered pairs of n-permutations with exactly k rise/falls or fall/rises, n >= 0, 0 <= k <= max{0,n-1}.

Original entry on oeis.org

1, 1, 2, 2, 10, 16, 10, 88, 200, 200, 88, 1216, 3536, 4896, 3536, 1216, 24176, 85872, 149152, 149152, 85872, 24176, 654424, 2743728, 5714472, 7176352, 5714472, 2743728, 654424, 23136128, 111842432, 270769536, 407103104, 407103104, 270769536, 111842432, 23136128

Offset: 0

Geoffrey Critzer, May 01 2023

Let ( (a_1,a_2,...,a_n), (b_1,b_2,...,b_n) ) be an ordered pair of n-permutations. Then the pairs (a_i,a_(i+1)) and (b_i,b_(i+1)) are both rises, both falls, a rise and a fall, or a fall and a rise. T(n,k) is the number of ordered pairs of n-permutations that have a total of k rise/falls and fall/rises.

			Triangle begins:
    1;
    1;
    2,    2;
   10,   16,   10;
   88,  200,  200,   88;
 1216, 3536, 4896, 3536, 1216;
 ...
In the ordered pair of permutations ( (1,2,3,5,4), (4,2,1,3,5) ) we have a rise/fall, rise/fall, rise/rise, fall/rise.  So this ordered pair is counted in T(5,3).

Alois P. Heinz, Rows n = 0..70, flattened
L. Carlitz, Richard Scoville, and Theresa Vaughan, Enumeration of pairs of permutations and sequences, Bull. Amer. Math. Soc. 80(5) (1974), 881-884.

Cf. A060350 (column k=0), A001044 (row sums), A259465.

b:= proc(n, u, v) option remember; expand(`if`(n=0, 1,
      add(add(b(n-1, u-j, v-i), i=1..v)+
          add(b(n-1, u-j, v+i-1)*x, i=1..n-v), j=1..u)+
      add(add(b(n-1, u+j-1, v-i)*x, i=1..v)+
          add(b(n-1, u+j-1, v+i-1), i=1..n-v), j=1..n-u)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0$2)):
seq(T(n), n=0..10);  # Alois P. Heinz, May 01 2023

nn = 8; A[z_] := Total[Select[Import["https://oeis.org/A060350/b060350.txt", "Table"],Length@# == 2 &][[All, 2]]*Table[z^n/n!^2, {n, 0, 250}]];B[n_] := n!^2; e[z_] := Sum[z^n/B[n], {n, 0, nn}]; Map[Select[#, # > 0 &] &,Table[B[n], {n, 0, nn}] CoefficientList[Series[((1 - u) A[(1 - u) z])/(1 - u A[(1 - u) z]), {z, 0, nn}], {z, u}]] // Flatten

Sum_{n>=0} Sum_{k=0..n-1} u^k*z^n/(n!)^2 = ((1 - u) A((1 - u) z))/(1 - u A((1 - u) z)) where A(z) = Sum_{n>=0} A060350*z^n/(n!)^2. Theorem 4 in Carlitz, Scoville, Vaughan link.

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A291902 Sums of the cubes of the descent set statistics for permutations on n elements.

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A291907 Numbers such that the nonzero digits in the base 3 expansion consists of two 1s and one 2.

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A137783 a(n) = the number of permutations (p(1), p(2), ..., p(2n+1)) of (1, 2, ..., 2n+1) where, for each k (2 <= k <= 2n+1), the sign of (p(k) - p(k-1)) equals the sign of (p(2n+2-k) - p(2n+3-k)).

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A362745 Triangular array read by rows. T(n,k) is the number of ordered pairs of n-permutations with exactly k rise/falls or fall/rises, n >= 0, 0 <= k <= max{0,n-1}.

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