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 A245182 Irregular triangle read by rows: T(n,k) (n>=2, 1<=k<=n) gives number of arrangements of the elements from the multiset M(n, 2) into exactly k disjoint cycles. 2

%I

%S 1,1,1,2,1,3,6,4,1,12,26,20,7,1,60,140,121,51,11,1,360,894,849,410,

%T 110,16,1,2520,6594,6763,3634,1135,211,22,1,20160,55152,60304,35336,

%U 12404,2723,371,29,1,181440,515808,595404,374940,144613,36001,5866,610,37,1

%N Irregular triangle read by rows: T(n,k) (n>=2, 1<=k<=n) gives number of arrangements of the elements from the multiset M(n, 2) into exactly k disjoint cycles.

%C The multiset M(n, 2) consists of 2 1's and one copy of each of 2..n-1. For example, M(5,2) is the multiset {1,1,2,3,4}. See Griffiths reference for formula. - _Andrew Howroyd_, Feb 24 2020

%H Martin Griffiths, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Griffiths/griffiths31.html">Generating Functions for Extended Stirling Numbers of the First Kind</a>, Journal of Integer Sequences, 17 (2014), #14.6.4.

%F T(n,k) = Sum_{m=0..k} |Stirling1(n-2, m)| * Sum_{j=0, 2-k+m} binomial(n+j-3, j) * A008284(2-j, k-m) for n >= 2. - _Andrew Howroyd_, Feb 24 2020

%e Triangle begins:

%e 1 1

%e 1 2 1

%e 3 6 4 1

%e 12 26 20 7 1

%e 60 140 121 51 11 1

%e 360 894 849 410 110 16 1

%e 2520 6594 6763 3634 1135 211 22 1

%e ...

%e From _Andrew Howroyd_, Feb 24 2020: (Start)

%e For n = 4, arrangements of the multiset {1,1,2,3} into cycles are:

%e T(4,1) = 3: (1123), (1132), (1213);

%e T(4,2) = 6: (112)(3), (113)(2), (123)(1), (132)(1), (11)(23) (12)(13);

%e T(4,3) = 4: (11)(2)(3), (12)(1)(3), (13)(1)(2), (23)(1)(1);

%e T(4,4) = 1: (1)(1)(2)(3).

%e (End)

%o T(n)={my(P=matrix(3, 3, n, k, #partitions(n-k, k-1))); matrix(n, n, n, k, if(n<2, 0, sum(m=0, k, abs(stirling(n-2, m, 1)) * sum(j=0, 2-k+m, binomial(n+j-3, j)*P[1+2-j, 1+k-m]))))}

%o {my(A=T(10)); for(n=2, #A, print(A[n, 1..n]))} \\ _Andrew Howroyd_, Feb 24 2020

%Y Cf. A008284, A245183, A245184.

%K nonn,tabf

%O 2,4

%A _N. J. A. Sloane_, Jul 17 2014

%E Terms a(36) and beyond from _Andrew Howroyd_, Feb 24 2020

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Last modified November 27 03:56 EST 2021. Contains 349345 sequences. (Running on oeis4.)