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Triangle read by rows: T(n,k) is the number of partial idempotent mappings (of an n-chain) with breadth exactly k.
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%I #23 Jan 08 2019 19:10:57

%S 1,1,1,1,2,3,1,3,9,10,1,4,18,40,41,1,5,30,100,205,196,1,6,45,200,615,

%T 1176,1057,1,7,63,350,1435,4116,7399,6322,1,8,84,560,2870,10976,29596,

%U 50576,41393,1,9,108,840,5166,24696,88788,227592,372537,293608

%N Triangle read by rows: T(n,k) is the number of partial idempotent mappings (of an n-chain) with breadth exactly k.

%D F. AlKharosi, W. AlNadabi and A. Umar, "Combinatorial results for idempotents in full and partial transformation semigroups", (submitted).

%H Haoliang Wang, Robert Simon, <a href="https://doi.org/10.1145/3267129.3267134">The Analysis of Synchronous All-to-All Communication Protocols for Wireless Systems</a>, Q2SWinet'18: Proceedings of the 14th ACM International Symposium on QoS and Security for Wireless and Mobile Networks (2018), 39-48.

%F T(n,k) = binomial(n,k) * Sum_{m=0..k} binomial(k,m)*m^(k-m).

%e T(3,2) = 9 because there are exactly 9 partial idempotent mappings (of a 3-chain) with breadth exactly 2, namely: (12-->11), (12-->22), (12-->12), (13-->11), (13-->33), (13-->13), (23-->22), (23-->33), (23-->23).

%e Triangle starts:

%e 1;

%e 1, 1;

%e 1, 2, 3;

%e 1, 3, 9, 10;

%e 1, 4, 18, 40, 41;

%e ...

%o (PARI) tabl(nn) = {for (n=0, nn, for (k=0, n, print1(binomial(n,k)*sum(m=0, k, binomial(k,m)*m^(k-m)), ", ");); print(););} \\ _Michel Marcus_, Jul 15 2015

%Y Cf. A000248, A052512.

%Y Row sums give A080108(n+1).

%K nonn,tabl

%O 0,5

%A _Wafa AlNadabi_, Jul 04 2015

%E More terms from _Michel Marcus_, Jul 15 2015