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A180389 Number of permutations of 1..n with number of rises (p(i+1)>p(i)) the same as number of rises in the inverse permutation. 3
1, 1, 2, 6, 22, 96, 492, 2952, 20588, 164990, 1497740, 15187692, 169974040, 2078905752, 27567259896, 393759207372, 6025346314756, 98317949671110, 1703879074519500, 31251488731748108, 604748393942784976, 12312387380060084768, 263079571362773145632 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Also equals sum of squares of the coefficients of the (numerators of) the G.F. for the count of monomials in the Schur polynomials of degree n (all partitions of weight n), in function of the number of variables v. - Wouter Meeussen, Dec 27 2010

Studied by Carlitz, Roselle, and Scoville in 'Permutations and Sequences with Repetitions by Number of Increases'. They refer to a rise/ascent as a 'jump', and consider the first entry of a permutation to always be a jump, so #jumps=#rises+1. Similarly, the number of rises in the inverse permutation/number of inverse descents corresponds to what they call the 'number of readings', and follow a convention so that #rises in inverse permutation+1=#readings. Formula can be attained by R_m(k,r), setting t=k, and summing k from 1 to m+1. - Kevin Dilks, Jun 09 2015

LINKS

Vaclav Kotesovec, Table of n, a(n) for n = 0..200

L. Carlitz et al., Permutations and sequences with repetitions by number of increases, J. Combin. Theory, 1 (1966), 350-374.

Wouter Meeussen, Schur Polynomials

Wouter Meeussen, Some notes on the link between A180389 and Schur polynomials

FORMULA

a(n) = Sum_{m=0..n+1} Sum_{i=0..m} Sum_{j=0..m} (-1)^{i+j} binomial(n+1,i) binomial(n+1,j) binomial((m-i)*(m-j)+n-1,n). - Kevin Dilks, Jun 09 2015

a(n) ~ sqrt(3) * n! / sqrt(Pi*n). - Vaclav Kotesovec, Jun 10 2015

EXAMPLE

For n=4, a(4)=22 are all permutations of length 4 except for 3142 (which has only one ascent, and two inverse ascents) and 2413 (which has two ascents, and only one inverse ascent). - Kevin Dilks, Jun 09 2015

MAPLE

seq(add(add(add((-1)^(i+j)*binomial(n+1, i)*binomial(n+1, j)*binomial((m-i)*(m-j)+n-1, n), i=0..m), j=0..m), m=0..n+1), n=0..30); # Kevin Dilks, Jun 09 2015

MATHEMATICA

Table[Sum[Sum[Sum[(-1)^(i+j)*Binomial[n+1, i]*Binomial[n+1, j]*Binomial[(m-i)*(m-j)+n-1, n], {i, 0, m}], {j, 0, m}], {m, 0, n+1}], {n, 0, 10}] (* Kevin Dilks, Jun 09 2015 *)

CROSSREFS

A180388(n) + a(n) = n! = A000142(n).

Sequence in context: A248836 A351919 A328500 * A177389 A130907 A054096

Adjacent sequences: A180386 A180387 A180388 * A180390 A180391 A180392

KEYWORD

nonn

AUTHOR

Leroy Quet, D. S. McNeil and R. H. Hardin in the Sequence Fans Mailing List Sep 01 2010

EXTENSIONS

a(15)-a(20) from Wouter Meeussen, Dec 27 2010

a(0)=1 prepended by Alois P. Heinz, Jun 10 2015

STATUS

approved

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Last modified November 27 05:12 EST 2022. Contains 358362 sequences. (Running on oeis4.)