A161492 Triangle T(n,m) read by rows: the number of skew Ferrers diagrams with area n and m columns.
1, 1, 1, 1, 2, 1, 1, 4, 3, 1, 1, 6, 8, 4, 1, 1, 9, 17, 13, 5, 1, 1, 12, 32, 34, 19, 6, 1, 1, 16, 55, 78, 58, 26, 7, 1, 1, 20, 89, 160, 154, 90, 34, 8, 1, 1, 25, 136, 305, 365, 269, 131, 43, 9, 1, 1, 30, 200, 544, 794, 716, 433, 182, 53, 10, 1, 1, 36, 284, 923, 1609, 1741, 1271, 657, 244, 64, 11, 1
Offset: 1
Examples
T(4,2)=4 counts the following 4 diagrams with area equal to 4 and 2 columns: .X..XX...X..XX XX..XX...X..X. X.......XX..X. From _Joerg Arndt_, Mar 23 2014: (Start) Triangle begins: 01: 1 02: 1 1 03: 1 2 1 04: 1 4 3 1 05: 1 6 8 4 1 06: 1 9 17 13 5 1 07: 1 12 32 34 19 6 1 08: 1 16 55 78 58 26 7 1 09: 1 20 89 160 154 90 34 8 1 10: 1 25 136 305 365 269 131 43 9 1 11: 1 30 200 544 794 716 433 182 53 10 1 12: 1 36 284 923 1609 1741 1271 657 ... (End)
Links
- Peter Bala, Illustration for the terms of row 4
- Peter Bala, Fountains of coins and skew Ferrers diagrams
- M. P. Delest and J. M. Fedou, Enumeration of skew Ferrers diagrams, Disc. Math. (1993) Vol.112, No. 1-3, pp. 65-79.
- Atli Fannar Franklín, Pattern avoidance enumerated by inversions, arXiv:2410.07467 [math.CO], 2024. See p. 19.
- Atli Fannar Franklín, Anders Claesson, Christian Bean, Henning Úlfarsson, and Jay Pantone, Restricted Permutations Enumerated by Inversions, arXiv:2406.16403 [cs.DM], 2024. See p. 5.
Programs
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Maple
qpoch := proc(a,q,n) mul( 1-a*q^k,k=0..n-1) ; end proc: A161492 := proc(n,m) local N,N2,ns ; N := 0 ; for ns from 0 to n+2 do N := N+ (-1)^ns *q^binomial(ns+1,2) / qpoch(q,q,ns) / qpoch(q,q,ns+1) *q^(ns+1) *t^(ns+1) ; N := taylor(N,q=0,n+1) ; end do: N2 := 0 ; for ns from 0 to n+2 do N2 := N2+ (-1)^ns*q^binomial(ns,2)/(qpoch(q,q,ns))^2*q^ns*t^ns ; N2 := taylor(N2,q=0,n+1) ; end do: coeftayl(N/N2,q=0,n) ; coeftayl(%,t=0,m) ; end proc: for a from 1 to 20 do for c from 1 to a do printf("%d ", A161492(a,c)) ; od: od: -
Mathematica
nmax = 13; qn[n_] := Product[1 - q^k, {k, 1, n}]; nm = Sum[(-1)^n q^(n(n+1)/2)/(qn[n] qn[n+1])(t q)^(n+1) + O[q]^nmax, {n, 0, nmax}]; dn = Sum[(-1)^n q^(n(n-1)/2)/(qn[n]^2)(t q)^n + O[q]^nmax, {n, 0, nmax}]; Rest[CoefficientList[#, t]]& /@ Rest[CoefficientList[nm/dn, q]] // Flatten (* Jean-François Alcover, Dec 19 2019, after Joerg Arndt *) -
PARI
/* formula from the Delest/Fedou reference: */ N=20; q='q+O('q^N); t='t; qn(n) = prod(k=1, n, 1-q^k ); nm = sum(n=0, N, (-1)^n* q^(n*(n+1)/2) / ( qn(n) * qn(n+1) ) * (t*q)^(n+1) ); dn = sum(n=0, N, (-1)^n* q^(n*(n-1)/2) / ( qn(n)^2 ) * (t*q)^n ); v=Vec(nm/dn); for(n=1,N-1,print(Vec(polrecip(Pol(v[n]))))); \\ print triangle \\ Joerg Arndt, Mar 23 2014
Formula
From Peter Bala, Jul 21 2019: (Start)
The following formulas all include an initial term T(0,0) = 1.
O.g.f. as a ratio of q-series: A(q,t) = N(q,t)/D(q,t) = 1 + q*t + q^2*(t + t^2) + q^3*(t + 2*t^2 + t^3) + ..., where N(q,t) = Sum_{n >= 0} (-1)^n*q^((n^2 + 3*n)/2)*t^n/Product_{k = 1..n} (1 - q^k)^2 and D(q,t) = Sum_{n >= 0} (-1)^n*q^((n^2 + n)/2)*t^n/Product_{k = 1..n} (1 - q^k)^2.
Continued fraction representations:
A(q,t) = 1/(1 - q*t/(1 - q/(1 - q^2*t/(1 - q^2/(1 - q^3*t/(1 - q^3/(1 - (...) ))))))).
A(q,t) = 1/(1 - q*t/(1 + q*(t - 1) - q*t/(1 + q*(t - q) - q*t/( 1 + q*(t - q^2) - q*t/( (...) ))))).
A(q,t) = 1/(1 - q*t - q^2*t/(1 - q*(1 + q*t) - q^4*t/(1 - q^2*(1 + q*t) - q^6*t/(1 - q^3*(1 + q*t) - q^8*t/( (...) ))))).
A(q,t) = 1/(1 - q*t - q^2*t/(1 - q^2*t - q/(1 - q^3*t - q^5*t/(1 - q^4*t - q^2/(1 - q^5*t - q^8*t/ (1 - q^6*t - q^3/(1 - q^7*t - q^11*t/(1 - q^8*t - (...) )))))))). (End)
Comments