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A213910 Irregular triangle read by rows: T(n,k) is the number of involutions of length n that have exactly k inversions; n>=0, 0<=k<=binomial(n,2). 2
1, 1, 1, 1, 1, 2, 0, 1, 1, 3, 1, 2, 1, 1, 1, 1, 4, 3, 3, 4, 2, 4, 1, 3, 0, 1, 1, 5, 6, 5, 9, 5, 10, 5, 9, 4, 7, 3, 3, 2, 1, 1, 1, 6, 10, 9, 16, 13, 19, 17, 19, 19, 17, 19, 13, 17, 7, 13, 3, 8, 1, 4, 0, 1, 1, 7, 15, 16, 26, 29, 34, 43, 39, 54, 41, 61, 40, 62, 36, 58, 28, 47, 21, 34, 15, 21, 10, 11, 6, 4, 3, 1, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,6

COMMENTS

Row sums are A000085.

Sum_{k>=0} T(n,k)*k = A211606(n).

Diagonal is A214086.

LINKS

Alois P. Heinz, Rows n = 0..40, flattened

FORMULA

T(n,k) = T(n-1,k) + Sum_{j=1..n-1} T(n-2,k-2*(n-j)+1) for n>=0, k>0; T(n,k) = 0 for n<0 or k<0; T(n,0) = 1 for n>=0. - Alois P. Heinz, Mar 07 2013

EXAMPLE

T(4,3) = 2 because we have: (3,2,1,4), (1,4,3,2).

Triangle T(n,k) begins:

  1;

  1;

  1, 1;

  1, 2, 0, 1;

  1, 3, 1, 2, 1, 1,  1;

  1, 4, 3, 3, 4, 2,  4, 1, 3, 0, 1;

  1, 5, 6, 5, 9, 5, 10, 5, 9, 4, 7, 3, 3, 2, 1, 1;

  ...

MAPLE

T:= proc(n) option remember; local f, g, j; if n<2 then 1 else

      f, g:= [T(n-1)], [T(n-2)]; for j to 2*n-3 by 2 do

      f:= zip((x, y)->x+y, f, [0$j, g[]], 0) od; f[] fi

    end:

seq(T(n), n=0..10);  # Alois P. Heinz, Mar 05 2013

MATHEMATICA

Table[Distribution[Map[Inversions, Involutions[n]], Range[0, Binomial[n, 2]]], {n, 0, 9}]//Flatten

CROSSREFS

Cf. A008302 (permutations of [n] with k inversions).

Sequence in context: A124035 A204184 A157897 * A288002 A140129 A029347

Adjacent sequences:  A213907 A213908 A213909 * A213911 A213912 A213913

KEYWORD

nonn,tabf

AUTHOR

Geoffrey Critzer, Mar 04 2013

STATUS

approved

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Last modified June 19 10:57 EDT 2021. Contains 345127 sequences. (Running on oeis4.)