A334060
Triangle read by rows: T(n,k) is the number of set partitions of {1..3n} into n sets of 3 with k disjoint strings of adjacent sets, each being a contiguous set of elements.
Original entry on oeis.org
1, 0, 1, 7, 3, 0, 219, 56, 5, 0, 12861, 2352, 183, 4, 0, 1215794, 174137, 11145, 323, 1, 0, 169509845, 19970411, 1078977, 30833, 334, 0, 0, 32774737463, 3280250014, 153076174, 4056764, 55379, 206, 0, 0, 8400108766161, 730845033406, 29989041076, 727278456, 10341101, 67730, 70, 0, 0
Offset: 0
Triangle begins:
1;
0, 1;
7, 3, 0;
219, 56, 5, 0;
12861, 2352, 183, 4, 0;
...
For n=2 and k=1 the configurations are (1,5,6),(2,3,4) and (1,2,6),(3,4,5) (i.e. configurations with a single contiguous set) and (1,2,3),(4,5,6) (i.e. two adjacent contiguous sets); hence T(2,1) = 3.
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CoefficientList[Normal[Series[Sum[y^j*(3*j)!/6^j/j!*((1-y*(1-z))/(1-y^2*(1-z)))^(3*j+1), {j, 0, 20}], {y, 0, 20}]], {y, z}]
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T(n)={my(v=Vec(sum(j=0, n, (3*j)! * x^j * (1-(1-y)*x + O(x*x^n))^(3*j+1) / (j! * 6^j * (1-(1-y)*x^2 + O(x*x^n))^(3*j+1))))); vector(#v, i, Vecrev(v[i], i))}
{ my(A=T(8)); for(n=1, #A, print(A[n])) }
A325754
Triangle read by rows giving the number of configurations of n indistinguishable pairs placed on the vertices of the ladder graph P_2 X P_n such that exactly k such pairs are joined by a horizontal edge.
Original entry on oeis.org
1, 1, 0, 2, 0, 1, 7, 4, 4, 0, 43, 38, 21, 2, 1, 372, 360, 168, 36, 9, 0, 4027, 3972, 1818, 478, 93, 6, 1, 51871, 51444, 23760, 6640, 1260, 144, 16, 0, 773186, 768732, 358723, 103154, 20205, 2734, 278, 12, 1, 13083385, 13027060, 6129670, 1796740, 363595, 52900, 5650, 400, 25, 0
Offset: 0
The first few rows of T(n,k) are:
1;
1, 0;
2, 0, 1;
7, 4, 4, 0;
43, 38, 21, 2, 1;
...
For n=2, let the vertex set of P_2 X P_2 be {A,B,C,D} and the edge set be {AB, AC, BD, CD}, where AB and CD are horizontal edges. For k=0, we may place the pairs on A, C and B, D or on A, D and B, C, hence T(2,0) = 2. If we place a pair on one of the horizontal edges we are forced to place the other pair on the remaining horizontal edge, hence T(2,1)=0 and T(2,2)=1.
Cf.
A046741,
A055140,
A079267 A178523,
A265167,
A318243,
A318244,
A318267,
A318268,
A318269,
A318270,
A325753.
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CoefficientList[Normal[Series[Sum[Factorial2[2*k-1]*y^k/(1-(1-z)*y)/(1+(1-z)*y)^(2*k+1), {k, 0, 20}], {y, 0, 20}]], {y, z}];
A334061
Triangle read by rows: T(n,k) is the number of set partitions of {1..4n} into n sets of 4 with k disjoint strings of adjacent sets, each being a contiguous set of elements.
Original entry on oeis.org
1, 0, 1, 31, 4, 0, 5474, 292, 9, 0, 2554091, 72318, 1206, 10, 0, 2502018819, 43707943, 438987, 2871, 5, 0, 4456194509950, 52717010017, 351487598, 1622954, 4355, 1, 0, 13077453070386914, 111615599664989, 528618296314, 1764575884, 4080889, 4385, 0, 0
Offset: 0
Triangle begins:
1;
0, 1;
31, 4, 0;
5474, 292, 9, 0;
2554091, 72318,1206, 10, 0;
...
For n=2 and k=1 the configurations are (1,6,7,8),(2,3,4,5), as well as (1,2,7,8),(3,4,5,6) and also (1,2,3,8),(4,5,6,7) (i.e. configurations with a single contiguous set) and (1,2,3,4),(5,6,7,8) (i.e. two adjacent contiguous sets); hence T(2,1) = 4.
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CoefficientList[Normal[Series[Sum[y^j*(4*j)!/24^j/j!*((1-y*(1-z))/(1-y^2*(1-z)))^(4*j+1), {j, 0, 20}], {y, 0, 20}]], {y, z}]
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T(n)={my(v=Vec(sum(j=0, n, (4*j)! * x^j * (1-(1-y)*x + O(x*x^n))^(4*j+1) / (j! * 24^j * (1-(1-y)*x^2 + O(x*x^n))^(4*j+1))))); vector(#v, i, Vecrev(v[i], i))}
{ my(A=T(8)); for(n=1, #A, print(A[n])) }
A367000
Triangle read by rows: T(n,k) is the total number of bubbles of size k found in linear chord diagrams on 2n vertices.
Original entry on oeis.org
0, 0, 2, 0, 0, 1, 8, 4, 2, 2, 0, 5, 42, 30, 20, 15, 12, 10, 0, 36, 300, 240, 186, 147, 120, 99, 82, 72, 0, 329, 2730, 2310, 1920, 1605, 1356, 1155, 988, 848, 730, 658, 0, 3655, 30240, 26460, 22890, 19845, 17280, 15105, 13242, 11634, 10240, 9027, 7968, 7310, 0, 47844
Offset: 0
The first few rows of T(n,k) are:
0, 0;
2, 0, 0, 1;
8, 4, 2, 2, 0, 5;
42, 30, 20, 15, 12, 10, 0, 36;
For n = 2, let the four vertices be A, B, C, D. The diagram consisting of the chords (A,B) and (C,D) has no bubbles. The diagram consisting of the chords (A,D) and (B,C) has two bubbles of size 1: The vertex A is one bubble and the vertex D is the other. The diagram consisting of the chords (A,C) and (B,D) is itself a bubble of size 4. Hence T(2,1) = 2 and T(2,4) = 1.
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N=2*n;
G=0; for(j=0,j=N/2, G=G+taylor((1/((1 + w*(-1 + w*y^2))^2))*((((w^2*y^2)/(2*(1 + w^2*y^2)^2))^j*(2*j)!/j!* (-1 + w)^2*(-1 + w*y^2)^2)/(1 + w^2*y^2) - ((y^2)/2)^j/j!*w*y^2*((-2 + 2*w + (3 -4*w)*w*y^2 + (w + 2*(-1 + w)*w^2)*y^4 + w^3*y^6 )*(2*j)!+(-y^4 + w*y^4+ w*y^6 - 2*w^2*y^6 + w^3*y^8 )*(2*j+2)!)),y,N+1); );
Tn=vector(N,x,0);
for(k=1,k=N,Tn[k]=polcoeff(polcoeff(G,N,y),k,w););
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