Wafa AlNadabi has authored 3 sequences.
A258579
Triangle read by rows: T(n,k) = number of partial idempotent mappings (of an n-chain) with (right) waist exactly k.
Original entry on oeis.org
1, 1, 1, 1, 2, 3, 1, 4, 7, 11, 1, 8, 17, 30, 48, 1, 16, 43, 86, 150, 241, 1, 32, 113, 258, 492, 846, 1358, 1, 64, 307, 806, 1686, 3108, 5276, 8445, 1, 128, 857, 2610, 6012, 11904, 21392, 35904, 57256, 1, 256, 2443, 8726, 22230, 47376, 90224, 158880, 263976, 419233
Offset: 0
T(3,2) = 7 because there are exactly 7 partial idempotent mappings (of a 3-chain) with right waist exactly 2, namely: (123-->222), (123-->122), (123-->121), (12-->22), (12-->12), (23-->22), (2-->2).
Triangle starts:
1;
1,1;
1,2,3;
1,4,7,11;
1,8,17,30,48;
...
- F. AlKharosi, W. AlNadabi and A. Umar, "Combinatorial results for idempotents in full and partial transformation semigroups", (submitted).
-
mybinom(x,y) = if ((x==-1) && (y==-1), 1, binomial(x,y));
tabl(nn) = {for (n=0, nn, for (k=0, n, print1(sum(m=0, k, mybinom(k-1, m-1) * (m+1)^(n-m)), ", "); ); print(); ); } \\ Michel Marcus, Jul 15 2015
A259759
Triangle read by rows: T(n,k) = number of partial idempotent mappings (of an n-chain) with collapse exactly k.
Original entry on oeis.org
1, 2, 0, 4, 0, 2, 8, 0, 12, 3, 16, 0, 48, 24, 16, 32, 0, 160, 120, 160, 65, 64, 0, 480, 480, 960, 780, 336, 128, 0, 1344, 1680, 4480, 5460, 4704, 1897, 256, 0, 3584, 5376, 17920, 29120, 37632, 30352, 11824, 512, 0, 9216, 16128, 64512, 131040, 225792, 273168, 212832, 80145
Offset: 0
T (3,2) = 12 because there are exactly 12 partial idempotent mappings (of a 3-chain) with collapse exactly 2, namely: (123-->113), (123-->121), (123-->122), (123-->223), (123-->133), (123--> 323), (12-->11), (12-->22), (23-->22), (23-->33), (13-->11), (13-->33).
Triangle starts:
1;
2,0;
4,0,2;
8,0,12,3;
16,0,48,24,16;
...
- F. AlKharosi, W. AlNadabi and A. Umar, "Combinatorial results for idempotents in full and partial transformation semigroups", (submitted).
-
tabl(nn) = {for (n=0, nn, for (k=0, n, print1(binomial(n,k)*sum(r=k, n, binomial(n-k,r-k)*sum(j=0, k, binomial(k,j)*stirling(k-j,j,2)*j!)), ", ");); print(););} \\ Michel Marcus, Jul 15 2015
A259760
Triangle read by rows: T(n,k) is the number of partial idempotent mappings (of an n-chain) with breadth exactly k.
Original entry on oeis.org
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, 1176, 1057, 1, 7, 63, 350, 1435, 4116, 7399, 6322, 1, 8, 84, 560, 2870, 10976, 29596, 50576, 41393, 1, 9, 108, 840, 5166, 24696, 88788, 227592, 372537, 293608
Offset: 0
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).
Triangle starts:
1;
1, 1;
1, 2, 3;
1, 3, 9, 10;
1, 4, 18, 40, 41;
...
- F. AlKharosi, W. AlNadabi and A. Umar, "Combinatorial results for idempotents in full and partial transformation semigroups", (submitted).
-
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