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 A177267 Triangle read by rows: T(n,k) is the number of permutations of {1,2,...,n} having genus k (see first comment for definition of genus). 7
 1, 2, 0, 5, 1, 0, 14, 10, 0, 0, 42, 70, 8, 0, 0, 132, 420, 168, 0, 0, 0, 429, 2310, 2121, 180, 0, 0, 0, 1430, 12012, 20790, 6088, 0, 0, 0, 0, 4862, 60060, 174174, 115720, 8064, 0, 0, 0, 0, 16796, 291720, 1309308, 1624480, 386496, 0, 0, 0, 0, 0, 58786, 1385670, 9087078, 18748730, 10031736, 604800, 0, 0, 0, 0, 0, 208012, 6466460, 59306676, 188208020, 186698512, 38113920, 0 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS The genus g(p) of a permutation p of {1,2,...,n} is defined by g(p)=(1/2)[n+1-z(p)-z(cp')], where p' is the inverse permutation of p, c = 234...n1 = (1,2,...,n), and z(q) is the number of cycles of the permutation q. The sum of the entries in row n is n!. The number of nonzero entries in row n is floor((n+1)/2). T(n,0) = A000108(n) (the Catalan numbers). Apparently T(n,1) = A002802(n-3). Last nonzero terms in rows with odd n appear to be A060593. [Joerg Arndt, Nov 01 2012.] REFERENCES S. Dulucq and R. Simion, Combinatorial statistics on alternating permutations, J. Algebraic Combinatorics, 8, 1998, 169-191. LINKS Alois P. Heinz, Rows n = 1..141, flattened FORMULA Let p(n, x) := g.f. of row n. Then (n+1) * p(n, x) = (4*n-2) * p(n-1, x) + x * (n-2) * (n-1)^2 * p(n-2, x). - Michael Somos, Sep 02 2017 EXAMPLE T(3,1)=1 because 312 is the only permutation of {1,2,3} with genus 1 (we have p=312=(132), cp'=231*231=312=(132) and so g(p)=(1/2)(3+1-1-1)=1). Triangle starts: [ 1]  1, [ 2]  2, 0, [ 3]  5, 1, 0, [ 4]  14, 10, 0, 0, [ 5]  42, 70, 8, 0, 0, [ 6]  132, 420, 168, 0, 0, 0, [ 7]  429, 2310, 2121, 180, 0, 0, 0, [ 8]  1430, 12012, 20790, 6088, 0, 0, 0, 0, [ 9]  4862, 60060, 174174, 115720, 8064, 0, 0, 0, 0, [10]  16796, 291720, 1309308, 1624480, 386496, 0, 0, 0, 0, 0, [11]  58786, 1385670, 9087078, 18748730, 10031736, 604800, 0, 0, ..., [12]  208012, 6466460, 59306676, 188208020, 186698512, 38113920, 0, ..., [13]  742900, 29745716, 368588220, 1700309468, 2788065280, 1271140416, 68428800, 0, ..., ... MAPLE n := 8: with(combinat): P := permute(n): inv := proc (p) local j, q: for j to nops(p) do q[p[j]] := j end do: [seq(q[i], i = 1 .. nops(p))] end proc: nrcyc := proc (p) local nrfp, pc: nrfp := proc (p) local ct, j: ct := 0: for j to nops(p) do if p[j] = j then ct := ct+1 else end if end do: ct end proc: pc := convert(p, disjcyc): nops(pc)+nrfp(p) end proc: b := proc (p) local c: c := [seq(i+1, i = 1 .. nops(p)-1), 1]: [seq(c[p[j]], j = 1 .. nops(p))] end proc: gen := proc (p) options operator, arrow: (1/2)*nops(p)+1/2-(1/2)*nrcyc(p)-(1/2)*nrcyc(b(inv(p))) end proc: f[n] := sort(add(t^gen(P[j]), j = 1 .. factorial(n))): seq(coeff(f[n], t, j), j = 0 .. ceil((1/2)*n)-1); # yields the entries in the specified row n MATHEMATICA T[ n_, k_] := If[ n < 1 || k >= n, 0, Module[{pn = Table[i, {i, n}]}, Do[ pn[[i]] = ((4 i - 2) pn[[i - 1]] + x (i - 2) (i - 1)^2 pn[[i - 2]])/(i + 1) // Expand, {i, 3, n}]; Coefficient[pn[[n]], x, k]]]; (* Michael Somos, Sep 02 2017 *) CROSSREFS Cf. A178514 (genus of derangements), A178515 (genus of involutions), A178516 (genus of up-down permutations), A178517 (genus of non-derangement permutations), A178518 (permutations of [n] having genus 0 and p(1)=k), A185209 (genus of connected permutations), A218538 (genus of permutations avoiding [x,x+1]). Cf. A000108, A002802. Sequence in context: A202209 A201730 A188449 * A188445 A327806 A319683 Adjacent sequences:  A177264 A177265 A177266 * A177268 A177269 A177270 KEYWORD nonn,tabl AUTHOR Emeric Deutsch, May 27 2010 EXTENSIONS Definition corrected by Emeric Deutsch, May 29 2010 Terms for rows 12 and 13 from Joerg Arndt, Jan 24 2011. STATUS approved

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Last modified February 19 01:00 EST 2020. Contains 332028 sequences. (Running on oeis4.)