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A262234 Number of ordered pairs (p,q) of permutations of [n] with equal up-down signatures and p(1)=q(1) if n>0. 5
1, 1, 2, 6, 36, 376, 6040, 137320, 4188320, 164844064, 8129435904, 490812707456, 35601674684160, 3054648986802432, 305980722047302144, 35383891435049320960, 4678354778386866393088, 701273672223926436540416, 118292442535368693562662912 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

Vaclav Kotesovec, Table of n, a(n) for n = 0..250 (terms 0..110 from Alois P. Heinz)

FORMULA

a(n) ~ c * d^n * n!^2 / n, where d = 0.552406011965766199179395470003589240257321... and c = 2.1899604476932970295731699544312... - Vaclav Kotesovec, Sep 18 2020

EXAMPLE

a(1) = 1: (1,1).

a(2) = 2: (12,12), (21,21).

a(3) = 6: (123,123), (132,132), (213,213), (231,231), (312,312), (321,321).

a(4) = 36: (1234,1234), (1243,1243), (1243,1342), (1324,1324), (1324,1423), (1342,1243), (1342,1342), (1423,1324), (1423,1423), (1432,1432), (2134,2134), (2143,2143), (2314,2314), (2314,2413), (2341,2341), (2413,2314), (2413,2413), (2431,2431), (3124,3124), (3142,3142), (3142,3241), (3214,3214), (3241,3142), (3241,3241), (3412,3412), (3421,3421), (4123,4123), (4132,4132), (4132,4231), (4213,4213), (4213,4312), (4231,4132), (4231,4231), (4312,4213), (4312,4312), (4321,4321).

MAPLE

b:= proc(u, o, h) option remember; `if`(u+o=0, 1,

      add(add(b(u-j, o+j-1, h+i-1), i=1..u+o-h), j=1..u)+

      add(add(b(u+j-1, o-j, h-i), i=1..h), j=1..o))

    end:

a:= n-> `if`(n=0, 1, add(b(j-1, n-j, n-j), j=1..n)):

seq(a(n), n=0..20);

MATHEMATICA

b[u_, o_, h_] := b[u, o, h] = If[u + o == 0, 1, Sum[Sum[b[u - j, o + j - 1, h + i - 1], {i, 1, u + o - h}], {j, 1, u}] + Sum[Sum[b[u + j - 1, o - j, h - i], {i, 1, h}], {j, 1, o}]];

a[n_] := If[n == 0, 1, Sum[b[j - 1, n - j, n - j], {j, 1, n}]];

a /@ Range[0, 20] (* Jean-Fran├žois Alcover, Dec 19 2020, after Alois P. Heinz *)

CROSSREFS

Row sums of A262372.

Cf. A060350, A262233, A262241.

Sequence in context: A133892 A196870 A089709 * A055512 A078973 A208650

Adjacent sequences:  A262231 A262232 A262233 * A262235 A262236 A262237

KEYWORD

nonn

AUTHOR

Alois P. Heinz, Sep 15 2015

STATUS

approved

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Last modified July 30 06:21 EDT 2021. Contains 346348 sequences. (Running on oeis4.)