Brant Jones has authored 5 sequences.
A129778
Number of Deodhar elements in the finite Weyl group D_n.
Original entry on oeis.org
2, 5, 14, 48, 167, 575, 1976, 6791
Offset: 1
Brant Jones (brant(AT)math.washington.edu), May 17 2007
a(4)=48 because there are 48 fully commutative elements in D_4 and since the first non-Deodhar fully-commutative element does not appear until D_6, these are all of the Deodhar elements in D_4.
- S. Billey and G. S. Warrington, Kazhdan-Lusztig polynomials for 321-hexagon-avoiding permutations, J. Algebraic Combin., 13(2):111-136, 2001.
- V. Deodhar, A combinatorial setting for questions in Kazhdan-Lusztig theory, Geom. Dedicata, 36(1): 95-119, 1990.
A129775
Number of maximally clustered permutations in S_n; the maximally clustered permutations are those that avoid 3421, 4312 and 4321.
Original entry on oeis.org
1, 1, 2, 6, 21, 78, 298, 1157, 4539, 17936, 71251, 284188, 1137076, 4561093, 18333337, 73816489, 297635750, 1201551286, 4855672249, 19640147061, 79501958895, 322037615290, 1305256267511, 5293166568270, 21475362822956, 87166344495561, 353933533606927
Offset: 0
Brant Jones (brant(AT)math.washington.edu), May 17 2007
a(5)=78 because there are 78 permutations of size 5 that avoid 3421, 4312 and 4321.
G.f. = 1 + x + 2*x^2 + 6*x^3 + 21*x^4 + 78*x^5 + 298*x^6 + 1157*x^7 + 4539*x^8 + ...
- Paul Barry, On a transformation of Riordan moment sequences, arXiv:1802.03443 [math.CO], 2018.
- David Callan, Toufik Mansour, and Mark Shattuck, Twelve subsets of permutations enumerated as maximally clustered permutations, Annales Mathematicae et Informaticae, 47 (2017) pp. 41-74.
- H. Denoncourt and B. Jones, The enumeration of maximally clustered permutations, arXiv:0704.3469 [math.CO], 2007-2008.
- S. B. Ekhad and M. Yang, Proofs of Linear Recurrences of Coefficients of Certain Algebraic Formal Power Series Conjectured in the On-Line Encyclopedia Of Integer Sequences, (2017)
- Jozsef Losonczy, Maximally clustered elements and Schubert varieties, Annals of Combinatorics 11 (2) (2007) 195-212.
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a[ n_] := SeriesCoefficient[ 1 + 2 x^2 / (-1 + 4 x - 2 x^2 + Sqrt[1 - 4 x]), {x, 0, n}]; (* Michael Somos, Jan 01 2014 *)
a[n_] := 1+Sum[(m Binomial[2(n-m), n-m-1] Hypergeometric2F1[m+1, m-n+1, n-m+2, -1])/(n-m), {m, 1, n-1}]; Table[a[n], {n, 0, 26}] (* Jean-François Alcover, Dec 14 2018, after Vladimir Kruchinin *)
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a(n):=if n=0 then 1 else sum(sum(k*binomial(m+k-1,m-1)*binomial(2*(n-m),n-m-k),k,1,n-m)/(n-m),m,1,n-1)+1; /* Vladimir Kruchinin, Oct 11 2011 */
A129777
Number of freely-braided hexagon-avoiding permutations in S_n; the freely-braided hexagon-avoiding permutations are those that avoid 3421, 4231, 4312, 4321, 46718235, 46781235, 56718234 and 56781234.
Original entry on oeis.org
1, 1, 2, 6, 20, 71, 260, 971, 3670, 13968, 53369, 204352, 783408, 3005284, 11533014, 44267854, 169935041, 652385639, 2504613713, 9615798516, 36917689075, 141737959416, 544175811783, 2089262741393, 8021347093432, 30796530585417, 118237818141689, 453953210838465
Offset: 0
Brant Jones (brant(AT)math.washington.edu), May 17 2007
a(8)=3670 because there are 3670 permutations of size 8 that avoid 3421, 4231, 4312, 4321, 46718235, 46781235, 56718234 and 56781234.
- Jozsef Losonczy, Maximally clustered elements and Schubert varieties, Ann. Comb. 11 (2007), no. 2, 195-212.
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LinearRecurrence[{6, -9, 3, 1, -8, -1, 1}, {1, 2, 6, 20, 71, 260, 971}, 27] (* Jean-François Alcover, Feb 02 2019 *)
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lista(nt) = { my(x = 'x + 'x*O('x^nt) ); P = (-x^7-2*x^6+2*x^5+x^4-3*x^3+4*x^2-x) / (x^7-x^6-8*x^5+x^4+3*x^3-9*x^2+6*x-1); print(Vec(P));} \\ Michel Marcus, Mar 17 2013
A129776
Number of maximally-clustered hexagon-avoiding permutations in S_n; the maximally-clustered hexagon-avoiding permutations are those that avoid 3421, 4312, 4321, 46718235, 46781235, 56718234, 56781234.
Original entry on oeis.org
1, 1, 2, 6, 21, 78, 298, 1157, 4535, 17872, 70644, 279704, 1108462, 4395045, 17431206, 69144643, 274300461, 1088215370, 4317321235, 17128527716, 67956202025, 269612504850, 1069675361622, 4243893926396, 16837490364983, 66802139457897, 265035151393777
Offset: 0
Brant Jones (brant(AT)math.washington.edu), May 17 2007
a(8)=4535 because there are 4535 permutations of size 8 that avoid 3421, 4312, 4321, 46718235, 46781235, 56718234 and 56781234.
- Jozsef Losonczy, Maximally clustered elements and Schubert varieties, Ann. Comb. 11 (2007), no. 2, 195-212.
A088960
Triangle read by rows: T(n,k) = number of configurations of k non-attacking bishops on the white squares of an n X n chessboard (for n even, 0 <= k < n).
Original entry on oeis.org
1, 2, 1, 8, 14, 4, 1, 18, 98, 184, 100, 8, 1, 32, 356, 1704, 3532, 2816, 632, 16, 1, 50, 940, 8480, 38932, 89256, 93800, 37600, 3856, 32, 1, 72, 2050, 29900, 242292, 1109184, 2800016, 3653280, 2180656, 474368, 23264, 64
Offset: 2
Brant Jones (brant(AT)math.washington.edu), Oct 28 2003
T(4,1) = 8 because there are 8 white squares on the 4 X 4 board to put one bishop; T(4,3) = 4 because we must place one bishop on each of three principal diagonal lines, which can be accomplished in 2*1*2=4 ways.
Triangle begins:
1, 2
1, 8, 14, 4
1, 18, 98, 184, 100, 8
1, 32, 356, 1704, 3532, 2816, 632, 16
- R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 1, 1997; see section 2.4.
- Vincenzo Librandi, Table of n, a(n) for n = 2..1000
- Irving Kaplansky and John Riordan, The problem of the rooks and its applications, Duke Mathematical Journal 13.2 (1946): 259-268. See Section 9.
- Irving Kaplansky and John Riordan, The problem of the rooks and its applications, in Combinatorics, Duke Mathematical Journal, 13.2 (1946): 259-268. See Section 9. [Annotated scanned copy]
- S.-M. Ma, T. Mansour, M. Schork. Normal ordering problem and the extensions of the Stirling grammar, arXiv preprint arXiv:1308.0169, 2013
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T[n_, k_] := (Sum[(-1)^j*Binomial[n - k - 1, j]/(n - k - 1)!*(n - k + 1 - j)^(n/2)*(n - k - j)^(n/2 - 1), {j, 0, n - k - 1}]); Flatten[Table[T[n, k], {n, 2, 12, 2}, {k, 0, n - 1}]] (* Vaclav Kotesovec, Mar 24 2011 *)
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